A GENERAL SO(3)-MONOPOLE COBORDISM FORMULA RELATING DONALDSON AND SEIBERG-WITTEN INVARIANTS

PAUL M. N. FEEHAN AND THOMAS G. LENESS

Abstract. We prove an analogue of the Kotschick-Morgan conjecture [19] in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. Although this article is in a preliminary form, it appears to be of sufficient interest that it is worth now distributing in its present shape.

1. Introduction

In this article, a sequel to [14, 15, 16, 9, 10], we complete the first step in the proof of the Witten and Moore-Witten formulas relating the Seiberg-Witten and Donaldson invariants [28], [22]. In particular, we prove that pairings of certain cohomology classes with the links of lower level Seiberg-Witten moduli subspaces of the Uhlenbeck compactification of the moduli space of SO(3) monopoles depends only on the Seiberg-Witten invariant and the homotopy type of a four-manifold X, through its Euler characteristic χ and signature σ (see Theorem 11.1). This leads to the following result. Theorem 1.1. Let X be a closed, connected, oriented smooth four-manifold with b_{1}(X) = 0, odd b_{2}^{+}(X) > 1. Let Λ, w ∈ H^{2}(X; ℤ) obey w − Λ ≡ w_{2}(X) (mod 2). Let δ, m be nonnegative integers for which m ≤ [δ/2], where [ ·] denotes the greatest integer function, and δ ≡ −w^{2} − \frac{3}{4}(χ + σ) (mod 4), with Λ and δ obeying δ < i(Λ), where i(Λ) = Λ^{2} − \frac{1}{4}(χ + σ). Then for any h ∈ H_{2}(X; ℝ) and generator x ∈ H_{0}(X; ℤ), we have the following expression for the Donaldson invariant:

(1.1)

D_{X}^{w}(h^{δ−2m}x^{m}) = ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+w·c_{1}(𝔰))}SW_{X}(𝔰)

× ∑_{i=0}^{min(ℓ,[δ/2]−m)} (p_{δ,ℓ,m,i}(c_{1}(𝔰) − Λ, Λ)Q_{X}^{i}) (h),

where Q_{X} is the intersection form on H_{2}(X; ℝ), ℓ = \frac{1}{4}(δ + (c_{1}(𝔰) − Λ)^{2} + \frac{3}{4}(χ + σ)) and p_{δ,ℓ,m,i}(·, ·) is a degree δ − 2m − 2i homogeneous polynomial with coefficients which are universal functions of

χ, σ, c_{1}(𝔰)^{2}, Λ^{2}, c_{1}(𝔰) · Λ, δ, m, ℓ.

Recall that for a ‘spin^{u} structure’ 𝔱 on X (see [15] or §2 here), the moduli space, M_{𝔱}, of SO(3) monopoles was a smooth manifold away from two types of singular subspaces. The first type of subspace was identified with the moduli space of anti-self-dual connections [15]. The second type of subspace, that of reducible SO(3) monopoles, was identified with the Seiberg-Witten moduli space M_{𝔰} where the spin^{u} structure 𝔱 admits a splitting 𝔱 = 𝔰⊕𝔰⊗L [15]. The Uhlenbeck compactification M̄_{𝔱}/S^{1} of M_{𝔱}/S^{1} provides a cobordism between a link of the moduli space of anti-self-dual connections and links, L̄_{𝔱,𝔰}, of subspaces of reducible SO(3) monopoles. However, additional reducible SO(3) monopoles, in the form of subspaces,

Date: This version: March 5, 2002. Preliminary version. math.DG/yymmnnn.

PMNF was supported in part by NSF grants DMS-9704174 and DMS-0125170. TGL was supported in part by NSF grant DMS-0103677.

(1.2)  M_{𝔰} × Sym^{ℓ}(X),

appear in the lower levels of the Uhlenbeck compactification. The intersection number of certain geometric representatives of cohomology classes with the link of the anti-self-dual connections yields a multiple of the Donaldson invariant. The main technical result of this paper, Theorem 11.1, is a formula (which is not completely explicit) for the intersection number of these geometric representatives with the link L̄_{𝔱,𝔰} of the subspace (1.2) of M̄_{𝔱}/S^{1}.

We note that a statement similar to Theorem 11.1 can also be proved by the methods of this article, without the assumptions that b_{1}(X) = 0 or z = h^{δ−2m}x^{m}, but the resulting expression becomes considerably more complicated. While the precise expression for the intersection number in (11.1) is still unknown, we also prove the ‘multiplicity conjecture’ in [16, 6, 11] (see Theorem 11.2 here), which holds even without the simplifying assumptions.

1.1. Outline of the argument. We have described the program for proving Witten’s conjecture relating the Donaldson and Seiberg-Witten invariants for smooth four-manifolds extensively elsewhere, [14, 15, 13, 11]. The main technical difficulty with this program has been computing the intersection pairings (11.1) when the Seiberg-Witten monopoles lie in a lower level of the Uhlenbeck compactification of the moduli space of SO(3) monopoles. Such a computation was completed in [11] in the case when the reducibles lie in the first level; that is, when the Seiberg-Witten stratum has the form M_{𝔰} × X. The computation behind Theorems 11.1 and 11.2 is a qualitative understanding of the intersection pairing for the general case: Seiberg-Witten strata of the form M_{𝔰} × Sym^{ℓ}(X). In this article, we give the topological constructions necessary to prove Theorem 11.1; the required analytic results are established in [9, 10].

The gluing theorems, [9, 10], describe neighborhoods of the strata, M_{𝔰}×Σ, of the Seiberg-Witten monopoles M_{𝔰}×Sym^{ℓ}(X), where Σ is a stratum of the symmetric product Sym^{ℓ}(X). Roughly speaking, there is a fiber bundle, M(Σ) → Gl(Σ) → M_{𝔰}×Σ, with a well-understood structure group and fiber M(Σ) given by a product of moduli spaces of anti-self-dual connections on S^{4}. There is a section of a pseudo-vector bundle Υ(Σ) → Gl(Σ) and the gluing map identifies the zero-locus of this section with a neighborhood of M_{𝔰} × Σ in the moduli space of SO(3) monopoles.

There are two main sources of the difficulty in computing the intersection pairings (11.1) when ℓ > 1. The first difficulty is the presence of higher charge moduli spaces of anti-self-dual connections on S^{4} in the fiber M(Σ). The topology of these spaces is in still unknown although the needed information for the case ℓ = 2 has been worked out in [21]. We do not address this problem here. Instead, we use a pushforward-pullback argument (see §11.2) to isolate the topology of these fibers into ‘universal constants’ of the type appearing in (11.1). The second difficulty, which we do address here, is the presence of more than one stratum Σ ⊂ Sym^{ℓ}(X), and the resulting need for more than one gluing map to parameterize a neighborhood of M_{𝔰} × Sym^{ℓ}(X).

1.1.1. The problem of overlaps. To compute the intersection pairing (11.1) when ℓ ≥ 2, we must understand the overlap of the images of the gluing maps for different strata Σ and Σ′. We approach this problem by first understanding the overlap of the images of the splicing maps for different strata. Recall from [9] that the gluing map is defined by a perturbation of a splicing map which is defined, roughly, by using partitions of unity to piece together pairs on X and on S^{4}. Essentially, it is easier to compare two different splicing maps because the splicing maps can be written down explicitly.

Given two strata Σ and Σ′ of Sym^{ℓ}(X), with Σ ⊂ cl Σ′, and splicing maps

γ′_{Σ′} : Gl(Σ′) → C̄_{𝔱}, γ′_{Σ} : Gl(Σ) → C̄_{𝔱},

we achieve the desired comparison by defining a space of ‘overlap data’, Gl(Σ, Σ′), and maps

ρ_{Σ,Σ′}^{u} : Gl(Σ, Σ′) → Gl(Σ′), ρ_{Σ,Σ′}^{d} : Gl(Σ, Σ′) → Gl(Σ),

such that the diagram

(1.3)

Gl(Σ, Σ′) ⟶^{ρ_{Σ,Σ′}^{u}} Gl(Σ′)
ρ_{Σ,Σ′}^{d} ↓  ^{γ}′_{Σ′} ↓
Gl(Σ) ⟶^{γ′_{Σ}} C_{𝔱}

commutes and such that

Im(γ′_{Σ′}) ∩ Im(γ′_{Σ}) = Im(γ′_{Σ′} ∘ ρ_{Σ,Σ′}^{u}) = Im(γ′_{Σ} ∘ ρ_{Σ,Σ′}^{d}).

The diagram (1.3) is used to define the space of global splicing data as a pushout of the spaces of local splicing data Gl(Σ). Once this is accomplished, the result of the argument is largely a formal accounting of choices of cohomology classes with compact support and a use of the familiar pullback-pushforward technique.

1.1.2. The commutativity of splicing maps. Since the majority of the article is taken up with establishing the diagram (1.3), we now describe in more detail how that is carried out. The two strata Σ and Σ′ are given by partitions of ℓ. If Σ is the lower stratum, that is, Σ ⊂ cl Σ′, then the partition defining Σ′ is a refinement of the partition ℓ = κ_{1} + ⋯ + κ_{r} defining Σ. Write κ_{i} = κ_{i,1} + ⋯ + κ_{i,r_{i}} for this refinement. The overlap data space, Gl(Σ, Σ′), is, approximately, defined by a fiber bundle, M(Σ′) → Gl(Σ, Σ′) → ν(Σ, Σ′), where ν(Σ, Σ′) → Σ can be thought of as the normal bundle of Σ in Σ′. The principal bundle underlying Gl(Σ, Σ′) → ν(Σ, Σ′) is defined by a pullback of the principal bundle defining Gl(Σ) by the projection map ν(Σ, Σ′) → ν(Σ). The ‘upwards’ overlap map, ρ_{Σ,Σ′}^{u}, is then defined by an isomorphism between the restriction of the bundle Gl(Σ′) to ν(Σ, Σ′) and Gl(Σ, Σ′). The ‘downwards’ overlap map, ρ_{Σ,Σ′}^{d}, is defined by taking the data given by the fiber M(Σ′) and the fiber of ν(Σ, Σ′) → Σ and using it to splice together connections on S^{4}. That is, the fiber of ν(Σ, Σ′) → Σ can be interpreted as sets of points in ℝ^{4}, while the fiber M(Σ′) is defined by connections A_{i,j} on S^{4} with instanton numbers κ_{i,j}. Splicing the connections A_{i,j} to the trivial connection on ℝ^{4} at the given points in ℝ^{4} yields connections A_{i} with instanton number κ_{i} and thus a point in the fiber M(Σ). The downwards overlap map is then defined by applying the above splicing procedure to the fiber of Gl(Σ, Σ′) → ν(Σ, Σ′) → Σ and then using the identification of the principal bundle underlying Gl(Σ, Σ′) with that of Gl(Σ).

The key technical results in proving the commutativity of diagram (1.3) appear in Lemma 5.11 and Proposition 5.12. These statements can be thought of as providing an associativity result for splicing maps on S^{4}. In a simple case, they assert that if the connection A′_{1} is obtained from splicing the connections A_{1,1} and A_{1,2} to the trivial connection Θ at the points x_{1} and x_{2} in ℝ^{4},

A′_{1} = Θ#_{x_{1}}A_{1,1}#_{x_{2}}A_{1,2},

then the connection obtained by splicing A′_{1} and A_{2} to the trivial connection at y_{1} and y_{2} is equal to the connection obtained by splicing A_{1,1}, A_{1,2}, and A_{2} to the trivial connection at y_{1} + x_{1}, y_{1} + x_{2}, and y_{2} respectively:

(1.4)  \begin{matrix}Θ#_{y_{1}} (Θ#_{x_{1}}A_{1,1}#_{x_{2}}A_{1,2}) #_{y_{2}}A_{2} \\ = Θ#_{y_{1}+x_{1}}A_{1,1}#_{y_{1}+x_{2}}A_{1,2}#_{y_{2}}A_{2}. \end{matrix}

(Of course, this result only holds for connections which are suitably concentrated relative to the separation of the points in ℝ^{4}.)

Proposition 5.12 allows us to define the ‘spliced-end moduli space’, a deformation of the moduli space of anti-self-dual connections on S^{4}. The spliced-end moduli space differs from the actual moduli space of anti-self-dual connections on S^{4} only near the strata where the background connection is trivial. A neighborhood of these strata in the actual moduli space is given by the gluing map while a neighborhood of these strata in the spliced-end moduli space is given by the splicing maps. Proposition 5.12 is used to ensure that the union of the images of these different splicing maps is a smoothly stratified space.

Replacing the actual moduli space with the spliced-ends moduli space allows us to prove that the diagram (1.3) commutes by using the results of Proposition 6.9, an analogue of Proposition 5.12 for splicing onto pairs on X. Because the ends of M(Σ) are replaced by the image of a splicing map, the commutativity follows from an expression of the form (1.4). We note that Proposition 5.12 relies on the triviality of the background connection Θ and the flatness of the metric on ℝ^{4}. Thus, Proposition 6.9 only gives the commutativity of the diagram (1.3) for ‘crude splicing maps’ γ′′_{Σ} which are defined by flattening the background connection and pair on a larger subspace than the standard splicing map. However, once the space of global splicing data is constructed using the crude splicing maps, it is a straightforward matter to construct (see §6.7) isotopies of these crude splicing maps to the standard splicing maps.

1.2. Outline of the article. Section 2 reviews notation and definitions from our previous articles in this sequence. In §3, we describe a stratification of Sym^{ℓ}(X), and define normal bundles of the strata of Sym^{ℓ}(X). In §4, we show that the projection maps of the normal bundles introduced in §3 satisfy Thom-Mather type equalities on the overlap of their images. In §5, we define the spliced-ends moduli space of anti-self-dual connections on S^{4} and this is used in §6 to define the space of global splicing data. An analogous construction of an obstruction bundle over the space of global splicing data is carried out in §7 and §8. The cohomological computations are carried out in §9 where we compute the pullbacks of the relevant cohomology classes to the space of global splicing data, in §10 where we define the link L̄_{𝔱,𝔰}, and in §11 where we actually carry out the needed computations. We discuss the application of the main result of this article, Equation (1.1), to the proof of Witten’s conjecture in a separate article [12].

Acknowledgments. The authors are extremely grateful to Tom Mrowka for his constant encouragement over the years and in particular to some very fruitful conversations where he urged us to develop the concept of ‘spliced-end moduli spaces’ of instantons on S^{4}. He explained this concept to us in March 1999, and proposed it as a key idea in this part of our work on Witten’s conjecture and the Kotschick-Morgan conjecture, with the latter following almost immediately from the method developed in the present article, modulo the requisite gluing theory [9, 10, 7, 8].

2. Preliminaries

We begin in §2.1 by recalling the definition of the moduli space of SO(3) monopoles and its basic properties [14, 5]. In §2.2 we describe the stratum of zero-section monopoles, or anti-self-dual connections. In §2.3 we discuss the strata of reducible, or Seiberg-Witten monopoles, together with their ‘virtual’ neighborhoods and normal bundles. In §2.4 we define the cohomology classes which will be paired with the links of the anti-self-dual and Seiberg-Witten moduli spaces. In §2.5 we review the definition of the Donaldson series. Lastly, in §2.6 we describe the basic relation between the pairings with links of the antiself-dual and Seiberg-Witten moduli spaces provided by the SO(3)-monopole cobordism.

2.1. The moduli space of SO(3) monopoles. Throughout this article, (X, g) will denote a closed, connected, oriented, smooth, Riemannian four-manifold.

2.1.1. Clifford modules. Let V be a Hermitian vector bundle over (X, g) and let ρ : T^{∗}X → End_{ℂ}(V) be a real-linear map satisfying

(2.1)  ρ(α)^{2} = −g(α, α)id_{V} and ρ(α)^{†} = −ρ(α), α ∈ C^{∞}(T^{∗}X).

The map ρ uniquely extends to a linear isomorphism, ρ : Λ^{•}(T^{∗}X) ⊗_{ℝ} ℂ → End_{ℂ}(V), and gives V the structure of a Hermitian Clifford module for the complex Clifford algebra ℂℓ(T^{∗}X). There is a splitting V = V^{+} ⊕ V^{−}, where V^{±} are the ∓1 eigenspaces of ρ(vol). A unitary connection A on V is spin if

(2.2)  [∇_{A}, ρ(α)] = ρ(∇α) on C^{∞}(V),

for any α ∈ C^{∞}(T^{∗}X), where ∇ is the Levi-Civita connection.

A Hermitian Clifford module 𝔰 = (ρ, W) is a spin^{c} structure when W has complex rank four; it defines a class

(2.3)  c_{1}(𝔰) = c_{1}(W^{+}),

and every class in H^{2}(X; ℤ) lifting the second Stiefel-Whitney class, w_{2}(X) ∈ H^{2}(X; ℤ/2ℤ), arises this way.

We call a Hermitian Clifford module 𝔱 = (ρ, V) a spin^{u} structure when V has complex rank eight. Recall that 𝔤_{V} ⊂ 𝔰𝔲(V) is the SO(3) subbundle given by the span of the sections of the bundle 𝔰𝔲(V) which commute with the action of ℂℓ(T^{∗}X) on V. We obtain a splitting

(2.4)  𝔰𝔲(V^{+}) ≅ ρ(Λ^{+}) ⊕ iρ(Λ^{+}) ⊗_{ℝ} 𝔤_{V} ⊕ 𝔤_{V} ,

and similarly for 𝔰𝔲(V^{−}). The fibers V_{x}^{+} define complex lines whose tensor-product square is det(V_{x}^{+}) and thus a complex line bundle over X,

(2.5)  det^{\frac{1}{2}} (V^{+}).

A spin^{u} structure 𝔱 thus defines classes,

(2.6)  c_{1}(𝔱) = \frac{1}{2}c_{1}(V^{+}), p_{1}(𝔱) = p_{1}(𝔤_{V}), and w_{2}(𝔱) = w_{2}(𝔤_{V}).

Given W, one has an isomorphism V ≅ W ⊗_{ℂ} E of Hermitian Clifford modules, where E is a rank-two Hermitian vector bundle [15, Lemma 2.3]; then

𝔤_{V} = 𝔰𝔲(E) and det^{\frac{1}{2}} (V^{+}) = det(W^{+}) ⊗_{ℂ} det(E).

2.1.2. SO(3) monopoles. We fix a smooth unitary connection A_{Λ} on the line bundle det^{\frac{1}{2}} (V^{+}), let k ≥ 2 be an integer, and let A_{𝔱} be the affine space of L_{k}^{2} spin connections on V which induce the connection 2A_{Λ} on det(V^{+}). If A is a spin connection on V then it defines an SO(3) connection Â on the subbundle 𝔤_{V} ⊂ 𝔰𝔲(V) [15, Lemma 2.5]; conversely, every SO(3) connection on 𝔤_{V} lifts to a unique spin connection on V inducing the connection 2A_{Λ} on det(V^{+}) [15, Lemma 2.11].

Let G_{𝔱} denote the group of L_{k+1}^{2} unitary automorphisms of V which commute with ℂℓ(T^{∗}X) and which have Clifford-determinant one (see [15, Definition 2.6]). Define

(2.7)  C̃_{𝔱}(X) = A_{𝔱}(X) × L_{k}^{2}(X, V^{+}) and C_{𝔱} = C̃_{𝔱}/G_{𝔱}.

The action of G_{𝔱} on V induces an adjoint action on End_{ℂ}(V), acting as the identity on ρ(Λ_{ℂ}^{•}) ⊂ End_{ℂ}(V) and inducing an adjoint action on 𝔤_{V} ⊂ End_{ℂ}(V) (see [15, Lemma 2.7]). The space C̃_{𝔱} and hence C_{𝔱} carry circle actions induced by scalar multiplication on V:

(2.8)  S^{1} × V → V, (e^{iθ}, Φ) ↦ e^{iθ}Φ.

Because this action commutes with that of G_{𝔱}, the action (2.8) also defines an action on C_{𝔱}. Note that −1 ∈ S^{1} acts trivially on C_{𝔱}. Let C_{𝔱}^{0} ⊂ C_{𝔱} be the subspace represented by pairs whose spinor components are not identically zero, let C_{𝔱}^{∗} ⊂ C_{𝔱} be the subspace represented by pairs where the induced SO(3) connections on 𝔤_{V} are irreducible, and let C_{𝔱}^{∗,0} = C_{𝔱}^{∗} ∩ C_{𝔱}^{0}.

We call a pair (A, Φ) in C̃_{𝔱} a SO(3) monopole if

(2.9)  \begin{aligned}ad^{−1}(F + ; Â) − τρ^{−1}(Φ ⊗ Φ^{∗})_{00} & = 0, \\ D_{A}Φ + ρ(ϑ) & = 0. \end{aligned}

Here, D_{A} = ρ ∘ ∇_{A} : C^{∞}(X, V^{+}) → C^{∞}(X, V^{−}) is the Dirac operator; the isomorphism ad : 𝔤_{V} → 𝔰𝔬(𝔤_{V}) identifies the self-dual curvature F + ; Â, a section of Λ^{+} ⊗ 𝔰𝔬(𝔤_{V}), with ad^{−1}(F + ; Â), a section of Λ^{+} ⊗ 𝔤_{V}; the section τ of GL(Λ^{+}) is a perturbation close to the identity; the perturbation ϑ is a complex one-form close to zero; Φ^{∗} ∈ Hom(V^{+}, ℂ) is the pointwise Hermitian dual ⟨·, Φ⟩ of Φ; and (Φ ⊗ Φ^{∗})_{00} is the component of the section Φ ⊗ Φ^{∗} of i𝔲(V^{+}) lying in ρ(Λ^{+}) ⊗ 𝔤_{V} with respect to the splitting 𝔲(V^{+}) = i\underline{ℝ} ⊕ 𝔰𝔲(V^{+}) and decomposition (2.4) of 𝔰𝔲(V^{+}).

Equation (2.9) is invariant under the action of G_{𝔱}. We let M_{𝔱} ⊂ C_{𝔱} be the subspace represented by pairs satisfying equation (2.9) and write

(2.10)  M_{𝔱}^{∗} = M_{𝔱} ∩ C_{𝔱}^{∗}, M_{𝔱}^{0} = M_{𝔱} ∩ C_{𝔱}^{0}, and M_{𝔱}^{∗,0} = M_{𝔱} ∩ C_{𝔱}^{∗,0}.

Since equation (2.9) is invariant under the circle action induced by scalar multiplication on V, the subspaces (2.10) of C_{𝔱} are also invariant under this action. Theorem 2.1. [5, Theorem 1.1], [27] Let X be a closed, oriented, smooth four-manifold and let V be a complex rank-eight, Hermitian vector bundle over X. Then for parameters (ρ, g, τ, ϑ), which are generic in the sense of [5], and 𝔱 = (ρ, V), the space M_{𝔱}^{∗,0} is a smooth manifold of the expected dimension,

(2.11)

dim M_{𝔱}^{∗,0} = d_{a}(𝔱) + 2n_{a}(𝔱), where d_{a}(𝔱) = −2p_{1}(𝔱) − \frac{3}{2}(χ + σ),
n_{a}(𝔱) = \frac{1}{4}(p_{1}(𝔱) + c_{1}(𝔱)^{2} − σ).

For the remainder of the article, we assume that the perturbation parameters in (2.9) are chosen as indicated in Theorem 2.1. For each non-negative integer ℓ, let 𝔱_{ℓ} = (ρ, V_{ℓ}), where

c_{1}(V_{ℓ}) = c_{1}(V), p_{1}(𝔤_{V_{ℓ}}) = p_{1}(𝔤_{V}) + 4ℓ, and w_{2}(𝔤_{V_{ℓ}}) = w_{2}(𝔤_{V}).

We let M̄_{𝔱} denote the closure of M_{𝔱} in the space of ideal monopoles,

(2.12)  IM_{𝔱} = ⨆_{ℓ=0}^{∞}(M_{𝔱_{ℓ}} × Sym^{ℓ}(X)),

with respect to an Uhlenbeck topology [14, Definition 4.19] and call the intersection of M̄_{𝔱} with M_{𝔱_{ℓ}} × Sym^{ℓ}(X) its ℓ-th level. Theorem 2.2. [14, Theorem 1.1] Let X be a Riemannian four-manifold with spin^{u} structure 𝔱. Then there is a positive integer N, depending at most on the curvature of the chosen unitary connection on det(V^{+}) together with p_{1}(𝔱), such that the Uhlenbeck closure M̄_{𝔱} of M_{𝔱} in ⊔_{ℓ=0}^{N}(M_{𝔱_{ℓ}} × Sym^{ℓ}(X)) is a second countable, compact, Hausdorff space. The space M̄_{𝔱} carries a continuous circle action, which restricts to the circle action defined on M_{𝔱_{ℓ}} on each level.

2.2. Stratum of anti-self-dual or zero-section solutions. From equation (2.9), we see that the stratum of M_{𝔱} represented by pairs with zero spinor is identified with

{A ∈ A_{𝔱} : F + ; Â = 0}/G_{𝔱} ≅ M_{κ}^{w}(X, g),

the moduli space of g-anti-self-dual connections on the SO(3) bundle 𝔤_{V}, where κ = −\frac{1}{4}p_{1}(𝔱) and w ≡ w_{2}(𝔱) (mod 2). For a generic Riemannian metric g, the space M_{κ}^{w}(X, g) is a smooth manifold of the expected dimension −2p_{1}(𝔱) − \frac{3}{2}(χ + σ) = d_{a}(𝔱).

As explained in [15, §3.4.1], it is desirable to choose w (mod 2) so as to exclude points in M̄_{𝔱} with associated flat SO(3) connections, so we have a disjoint union,

(2.13)  M̄_{𝔱} ≅ M̄_{𝔱}^{∗,0} ⊔ M̄_{κ}^{w} ⊔ M̄_{𝔱}^{red},

where M̄_{𝔱}^{∗} ⊂ M̄_{𝔱} is the subspace represented by triples whose associated SO(3) connections are irreducible, M̄_{𝔱}^{0} ⊂ M̄_{𝔱} is the subspace represented by triples whose spinors are not identically zero, M̄_{𝔱}^{∗,0} = M̄_{𝔱}^{∗}∩M̄_{𝔱}^{0}, while M̄_{𝔱}^{red} ⊂ M̄_{𝔱} is the subspace M̄_{𝔱}−M̄_{𝔱}^{∗} represented by triples whose associated SO(3) connections are reducible. We recall the Definition 2.3. [16, Definition 3.20] A class v ∈ H^{2}(X; ℤ/2) is good if no integral lift of v is torsion.

If w (mod 2) is good, then the union (2.13) is disjoint, as desired. In practice, rather than constraining w (mod 2) itself, we replace X by the blow-up X#\overline{ℂℙ}^{2} and w by w + PD[e] (where e ∈ H_{2}(X; ℤ) is the exceptional class), noting that w + PD[e] (mod 2) is always good, and define gauge-theoretic invariants of X in terms of moduli spaces on X#\overline{ℂℙ}^{2}. When w (mod 2) is good, we define [15, Definition 3.7] the link of M̄_{κ}^{w} in M̄_{𝔱}/S^{1} by

(2.14)  L_{𝔱,κ}^{w} = {[A, Φ, x] ∈ M̄_{𝔱}/S^{1} : ∥Φ∥_{L^{2}}^{2} = ε},

where ε is a small positive constant; for generic ε, the link L_{𝔱,κ}^{w} is a smoothly-stratified, codimension-one subspace of M̄_{𝔱}/S^{1}.

2.3. Strata of Seiberg-Witten or reducible solutions. We call a pair (A, Φ) ∈ C̃_{𝔱} reducible if the connection A on V respects a splitting,

(2.15)  V = W ⊕ W ⊗ L = W ⊗ (\underline{ℂ} ⊕ L),

for some spin^{c} structure 𝔰 = (ρ, W) and complex line bundle L, in which case c_{1}(L) = c_{1}(𝔱) − c_{1}(𝔰). A spin connection A on V is reducible with respect to the splitting (2.15) if and only if Â is reducible with respect to the splitting 𝔤_{V} ≅ \underline{ℝ} ⊕ L, [15, Lemma 2.9]. If A is reducible, we can write A = B ⊕ B ⊗ A_{L}, where B is a spin connection on W and A_{L} is a unitary connection on L; then Â = d_{ℝ} ⊕ A_{L} and A_{L} = A_{Λ} ⊗ (B^{det})^{∗}, where B^{det} is the connection on det(W^{+}) induced by B on W.

2.3.1. Seiberg-Witten monopoles. Given a spin^{c} structure 𝔰 = (ρ, W) on X, let A_{𝔰} denote the affine space of L_{k}^{2} spin connections on W. Let G_{𝔰} denote the group of L_{k+1}^{2} unitary automorphisms of W, commuting with ℂℓ(T^{∗}X), which we identify with L_{k+1}^{2}(X, S^{1}). We then define

(2.16)  C̃_{𝔰} = A_{𝔰} × L_{k}^{2}(W^{+}) and C_{𝔰} = C̃_{𝔰}/G_{𝔰},

where G_{𝔰} acts on C̃_{𝔰} by

(2.17)  (s, (B, Ψ)) ↦ s(B, Ψ) = (B − (s^{−1}ds)id_{W} , sΨ).

We call a pair (B, Ψ) ∈ C̃_{𝔰} a Seiberg-Witten monopole if

(2.18)  \begin{aligned}Tr(F_{B}^{+}) − τρ^{−1}(Ψ ⊗ Ψ^{∗})_{0} − F^{+}(A_{Λ}) & = 0, \\ D_{B}Ψ + ρ(ϑ)Ψ & = 0, \end{aligned}

where Tr : 𝔲(W^{+}) → iℝ is defined by the trace on 2 × 2 complex matrices, (Ψ ⊗ Ψ^{∗})_{0} is the component of the section Ψ ⊗ Ψ^{∗} of i𝔲(W^{+}) contained in i𝔰𝔲(W^{+}), D_{B} : C^{∞}(W^{+}) → C^{∞}(W^{−}) is the Dirac operator, and A_{Λ} is a unitary connection on a line bundle with first Chern class Λ ∈ H^{2}(X; ℤ). The perturbations are chosen so that solutions to equation (2.18) are identified with reducible solutions to (2.9) when c_{1}(𝔱) = Λ. Let M̃_{𝔰} ⊂ C̃_{𝔰} be the subspace cut out by equation (2.18) and denote the moduli space of Seiberg-Witten monopoles by M_{𝔰} = M̃_{𝔰}/G_{𝔰}.

2.3.2. Seiberg-Witten invariants. We let C_{𝔰}^{0} ⊂ C_{𝔰} be the open subspace represented by pairs whose spinor components which are not identically zero and define a complex line bundle over C_{𝔰}^{0} × X by

(2.19)  𝕃_{𝔰} = C̃_{𝔰}^{0} ×_{G_{𝔰}} \underline{ℂ},

where \underline{ℂ} = X × ℂ and s ∈ G_{𝔰} acts on (B, Ψ) ∈ C̃_{𝔰} and (x, ζ) ∈ \underline{ℂ} by

(2.20)  ((B, Ψ), (x, ζ)) ↦ (s(B, Ψ), (x, s(x)^{−1}ζ)).

Define

(2.21)  𝔸_{2}(X) = Sym (H_{0}(X; ℝ)) ⊗ Λ^{•}(H_{1}(X; ℝ))

be the graded algebra, with z = β_{1}β_{2} ⋯ β_{r} having total degree deg(z) = ∑_{p}(2 − i_{p}), when β_{p} ∈ H_{i_{p}}(X; ℝ). Then the map,

μ_{𝔰} : H_{•}(X; ℝ) → H^{2−•}(C_{𝔰}^{0}; ℝ), β ↦ c_{1}(LL_{𝔰})/β,

extends in the usual way to a homomorphism of graded real algebras,

μ_{𝔰} : 𝔸_{2}(X) → H^{•}(C_{𝔰}^{0}; ℝ).

If x ∈ H_{0}(X; ℤ) denotes the positive generator, we set

(2.22)  μ_{𝔰} = c_{1}(𝕃_{𝔰})/x ∈ H^{2}(C_{𝔰}^{0}; ℤ).

Equivalently, μ_{𝔰} is the first Chern class of the S^{1} base-point fibration over C_{𝔰}^{0}. If b_{1}(X) = 0, then c_{1}(𝕃_{𝔰}) = μ_{𝔰} × 1 by [15, Lemma 2.14].

For b_{2}^{+}(X) > 0 and generic Riemannian metrics on X, the space M_{𝔰} contains zero-section pairs if and only if c_{1}(𝔰) − Λ is a torsion class by [23, Proposition 6.3.1]. If M_{𝔰} contains no zero-section pairs then, for generic perturbations, it is a compact, oriented, smooth manifold of dimension

(2.23)  d_{s}(𝔰) = dim M_{𝔰} = \frac{1}{4}(c_{1}(𝔰)^{2} − 2χ − 3σ).

Let X̃ = X#\overline{ℂℙ}^{2} denote the blow-up of X with exceptional class e ∈ H_{2}(X̃; ℤ) and denote its Poincaré dual by PD[e] ∈ H^{2}(X̃; ℤ). Let 𝔰^{±} = (ρ̃, W̃) denote the spin^{c} structure on X̃ with c_{1}(𝔰^{±}) = c_{1}(𝔰) ± PD[e] obtained by splicing the spin^{c} structure 𝔰 = (ρ, W) on X with the spin^{c} structure on \overline{ℂℙ}^{2} with first Chern class ± PD[e]. (See [16, §4.5] for an explanation of the relation between spin^{c} structures on X and X̃.) Now

c_{1}(𝔰) ± PD[e] − Λ ∈ H^{2}(X̃; ℤ)

is not a torsion class and so — for b_{2}^{+}(X) > 0, generic Riemannian metrics on X and related metrics on the connected sum X̃ — the moduli spaces M_{𝔰^{±}}(X̃) contain no zerosection pairs. Thus, for our choice of generic perturbations, the moduli spaces M_{𝔰^{±}}(X̃) are compact, oriented, smooth manifolds, both of dimension dim M_{𝔰}(X).

For b_{1}(X) = 0 and odd b_{2}^{+}(X) > 1, we define the Seiberg-Witten invariant by [16, §4.1]

(2.24)  SW_{X}(𝔰) = ⟨μ_{𝔰^{+}}^{d}, [M_{𝔰^{+}}(X̃)]⟩ = ⟨μ_{𝔰^{−}}^{d}, [M_{𝔰^{−}}(X̃)]⟩,

where 2d = d_{s}(𝔰) = d_{s}(𝔰^{±}). When b_{2}^{+}(X) = 1 the pairing on the right-hand side of definition (2.24) depends on the chamber in the positive cone of H^{2}(X̃; ℝ) determined by the period point of the Riemannian metric on X̃. The definition of the Seiberg-Witten invariant for this case is also given in [16, §4.1]: we assume that the class w_{2}(X) − Λ (mod 2) is good to avoid technical difficulties involved in chamber specification. Since w ≡ w_{2}(X) − Λ (mod 2), this coincides with the constraint we use to define the Donaldson invariants in §2.5 when b_{2}^{+}(X) = 1. We refer to [16, Lemma 4.1 & Remark 4.2] for a comparison of the chamber structures required for the definition of Donaldson and Seiberg-Witten invariants when b_{2}^{+}(X) = 1.

2.3.3. Reducible SO(3) monopoles. If 𝔱 = (ρ, V) and 𝔰 = (ρ, W) with V = W ⊕ W ⊗ L, then there is an embedding

(2.25)  ι : C̃_{𝔰} ↪ C̃_{𝔱}, (B, Ψ) ↦ (B ⊕ B ⊗ A_{Λ} ⊗ B^{det,∗}, Ψ ⊕ 0),

which is gauge-equivariant with respect to the homomorphism

(2.26)  ϱ : G_{𝔰} ↪ G_{𝔱}, s ↦ id_{W} ⊗ (\begin{matrix}s & 0 \\ 0 & s^{−1} \end{matrix}) .

According to [15, Lemma 3.13], the map (2.25) defines a topological embedding ι : M_{𝔰}^{0} ↪ M_{𝔱}, where M_{𝔰}^{0} = M_{𝔰} ∩ C_{𝔰}^{0} and an embedding of M_{𝔰} if w_{2}(𝔱) ≠ 0 or b_{1}(X) = 0; its image in M_{𝔱} is represented by pairs which are reducible with respect to the splitting V = W ⊕W ⊗L. Henceforth, we shall not distinguish between M_{𝔰} and its image in M_{𝔱} under this embedding.

2.3.4. Circle actions. When V = W ⊕ W ⊗ L and 𝔱 = (ρ, V), the space C̃_{𝔱} inherits a circle action defined by

(2.27)  S^{1} × V → V, (e^{iθ}, Ψ ⊕ Ψ′) ↦ Ψ ⊕ e^{iθ}Ψ′,

where Ψ ∈ C^{∞}(W) and Ψ′ ∈ C^{∞}(W ⊗ L). With respect to the splitting V = W ⊕ W ⊗ L, the actions (2.27) and (2.8) are related by

(2.28)  (\begin{matrix}1 & 0 \\ 0 & e^{i2θ} \end{matrix}) = e^{iθ}u, where u = (\begin{matrix}e^{−iθ} & 0 \\ 0 & e^{iθ} \end{matrix}) ∈ G_{𝔱},

and so, when we pass to the induced circle actions on the quotient C_{𝔱} = C̃_{𝔱}/G_{𝔱}, the actions (2.27) and (2.8) on C_{𝔱} differ only in their multiplicity. Recall [15, Lemma 3.11] that the image in C̃_{𝔱} of the map (2.25) contains all pairs which are fixed by the circle action (2.27).

2.3.5. The virtual normal bundle of the Seiberg-Witten moduli space. Suppose 𝔱 = (ρ, V) and 𝔰 = (ρ, W), with V = W ⊕ W ⊗ L, so we have a topological embedding M_{𝔰} ↪ M_{𝔱}; we assume M_{𝔰} contains no zero-section monopoles. Recall from [15, §3.5] that there exist finite-rank, complex vector bundles,

(2.29)  π_{Ξ} : Ξ_{𝔱,𝔰} → M_{𝔰} and π_{N} : N_{𝔱,𝔰} → M_{𝔰},

with Ξ_{𝔱,𝔰} ≅ M_{𝔰} × ℂ^{r_{Ξ}}, called the obstruction bundles and ambient normal bundles of M_{𝔰} ↪ M_{𝔱}, respectively. For a small enough positive radius ε, there are a topological embedding [15, Theorem 3.21] of an open tubular neighborhood,

(2.30)  γ_{𝔰} : N_{𝔱,𝔰}(ε) ↪ C_{𝔱},

and a smooth section χ_{𝔰} of the pulled-back complex vector bundle,

(2.31)  π_{N}^{∗}Ξ_{𝔱,𝔰} → N_{𝔱,𝔰}(ε),

such that the tubular map yields a homeomorphism

(2.32)  γ_{𝔰} : χ_{𝔰}^{−1}(0) ∩ N_{𝔱,𝔰}(ε) ≅ M_{𝔱} ∩ γ_{𝔰}(N_{𝔱,𝔰}(ε)),

restricting to a diffeomorphism on the complement of M_{𝔰} and identifying M_{𝔰} with its image ι(M_{𝔰}) ⊂ M_{𝔱}. The image γ_{𝔰}(N_{𝔱,𝔰}(ε)) is an open subset of an ambient moduli space, M_{𝔱,𝔰}^{vir} ⊂ C_{𝔱}.

Our terminology is loosely motivated by that of [17] and [25], where the goal (translated to our setting) would be to construct a virtual fundamental class for M_{𝔱}, given by the cap product of the fundamental class of an ambient space containing M_{𝔱} with the Euler class of a vector bundle over this ambient space, where M_{𝔱} is the zero locus of a (possibly nontransversally vanishing) section. Here, M_{𝔱,𝔰}^{vir} plays the role of the ambient space and (the pushforward of) Ξ_{𝔱,𝔰} the vector bundle with zero section yielding (an open neighborhood in) M_{𝔱}. Then, N_{𝔱,𝔰} is the normal bundle of M_{𝔰} ↪ M_{𝔱,𝔰}^{vir}, while [N_{𝔱,𝔰}] − [Ξ_{𝔱,𝔰}] would more properly be called the ‘virtual normal bundle’ of M_{𝔰} ↪ M_{𝔱}, in the language of K-theory.

Recall that minus the index of the SO(3)-monopole elliptic deformation complex [15, Equations (2.47) & (3.35)] at a reducible solution can be written as

(2.33)  dim M_{𝔱} = 2n_{s}(𝔱, 𝔰) + d_{s}(𝔰),

where d_{s}(𝔰) is the expected dimension of the Seiberg-Witten moduli space M_{𝔰} (see equation (2.23)), while n_{s}(𝔱, 𝔰) = n′_{s}(𝔱, 𝔰) + n′′_{s}(𝔱, 𝔰) is minus the complex index of the normal deformation operator [15, Equations (3.71) & (3.72)]), with

(2.34)  \begin{aligned}n′_{s}(𝔱, 𝔰) & = −(c_{1}(𝔱) − c_{1}(𝔰))^{2} − \frac{1}{2}(χ + σ), \\ n′′_{s}(𝔱, 𝔰) & = \frac{1}{8}(c_{1}(𝔰) − 2c_{1}(𝔱))^{2} − σ). \end{aligned}

If r_{Ξ} is the complex rank of Ξ_{𝔱,𝔰} → M_{𝔰}, and r_{N}(𝔱, 𝔰) is the complex rank of N_{𝔱,𝔰} → M_{𝔰}, then

(2.35)  r_{N}(𝔱, 𝔰) = n_{s}(𝔱, 𝔰) + r_{Ξ},

as we can see from the dimension relation (2.33) and the topological model (2.32).

The map (2.30) is circle equivariant when the circle acts trivially on M_{𝔰}, by scalar multiplication on the fibers of N_{𝔱,𝔰}(ε), and by the action (2.27) on C_{𝔱}. The bundle (2.31) and section χ_{𝔰} are circle equivariant if the circle acts on N_{𝔱,𝔰} and the fibers of γ_{𝔰}^{∗}Ξ_{𝔱,𝔰} by scalar multiplication.

Let Ñ_{𝔱,𝔰} → M̃_{𝔰} be the pullback of N_{𝔱,𝔰} by the projection M̃_{𝔰} → M_{𝔰} = M̃_{𝔰}/G_{𝔰}, so Ñ_{𝔱,𝔰} is a G_{𝔰}-equivariant bundle, where G_{𝔰} acts on the base M̃_{𝔰} by the usual gauge group action (2.17) and the induced action on the total space,

(2.36)  Ñ_{𝔱,𝔰} ⊂ M̃_{𝔰} × L_{k}^{2}(Λ^{1} ⊗_{ℝ} L) ⊕ L^{2}(W^{+} ⊗ L) ⊂ M̃_{𝔰} × L_{k}^{2}(Λ^{1} ⊗_{ℝ} 𝔤_{V}) ⊕ L^{2}(V^{+}),

via the embedding (2.26) of G_{𝔰} into G_{𝔱} and the splittings 𝔤_{V} ≅ \underline{ℝ} ⊕ L [15, Lemma 3.10] and V = W ⊕ W ⊗ L. Thus, s ∈ G_{𝔰} acts by scalar multiplication by s^{−2} on sections of Λ^{1} ⊗_{ℝ} L and by s^{−1} on sections of W^{+} ⊗ L [15, §3.5.4].

For a small enough positive ε, there is a smooth embedding [15, §3.5.4] of the open tubular neighborhood Ñ_{𝔱,𝔰}(ε),

(2.37)  γ̃_{𝔰} : Ñ_{𝔱,𝔰}(ε) → C̃_{𝔱},

which is gauge equivariant with respect to the preceding action of G_{𝔰}, and covers the topological embedding (2.30). The map (2.37) is circle equivariant, where the circle acts trivially on M̃_{𝔰}, by scalar multiplication on the fibers of Ñ_{𝔱,𝔰}(ε), and by the action (2.27) on C_{𝔱}. We note that the map (2.37) is also circle equivariant with respect to the action (2.8) on C_{𝔱}, if the circle acts on Ñ_{𝔱,𝔰} by

(2.38)  (e^{iθ}, (B, Ψ, β, ψ)) ↦ ϱ(e^{iθ})(B, Ψ, e^{2iθ}β, e^{2iθ}ψ) = (B, e^{iθ}Ψ, β, e^{iθ}ψ),

where (B, Ψ) ∈ M̃_{𝔰}, (β, ψ) ∈ L_{k}^{2}(Λ^{1} ⊗_{ℝ} L) ⊕ L^{2}(W^{+} ⊗ L), so (B, Ψ, β, ψ) ∈ Ñ_{𝔱,𝔰}, and ϱ : G_{𝔰} → G_{𝔱} is the homomorphism (2.26). This equivariance follows from the relation (2.28) between the actions (2.27) and (2.8).

We note that the Chern character of the bundle N_{𝔱,𝔰} is computed in [15, Theorem 3.29] while the Segre classes of this bundle are computed, under some additional assumptions, in [16, Lemma 4.11].

2.4. Cohomology classes on the moduli space of SO(3) monopoles. The identity (1.1) arises as an equality of pairings of suitable cohomology classes with a link in M̄_{𝔱}^{∗,0}/S^{1} of the anti-self-dual moduli subspace and the links of the Seiberg-Witten moduli subspaces in M̄_{𝔱}/S^{1}. We recall the definitions of these cohomology classes and their dual geometric representatives given in [16, §3].

The first kind of cohomology class is defined on M_{𝔱}^{∗}/S^{1}, via the associated SO(3) bundle,

(2.39)  𝔽_{𝔱} = C̃_{𝔱}^{∗}/S^{1} ×_{G_{𝔱}} 𝔤_{V} → C_{𝔱}^{∗}/S^{1} × X.

The group G_{𝔱} acts diagonally in (2.39), with G_{𝔱} acting on the left on 𝔤_{V}. We define [16, §3.1]

(2.40)  μ_{p} : H_{•}(X; ℝ) → H^{4−•}(C_{𝔱}^{∗}/S^{1}; ℝ), β ↦ −\frac{1}{4}p_{1}(𝔽_{𝔱})/β.

On restriction to M_{κ}^{w} ↪ M_{𝔱}, the cohomology classes μ_{p}(β) coincide with those used in the definition of Donaldson invariants [16, Lemma 3.1]. Define

(2.41)  𝔸(X) = Sym (H_{even}(X; ℝ)) ⊗ Λ^{•}(H_{odd}(X; ℝ))

to be the graded algebra, with z = β_{1}β_{2} ⋯ β_{r} having total degree deg(z) = ∑_{p}(4 − i_{p}), when β_{p} ∈ H_{i_{p}}(X; ℝ). Then μ_{p} extends in the usual way to a homomorphism of graded real algebras,

μ_{p} : 𝔸(X) → H^{•}(C_{𝔱}^{∗}/S^{1}; ℝ),

which preserves degrees. Next, we define a complex line bundle over C_{𝔱}^{∗,0}/S^{1},

(2.42)  𝕃_{𝔱} = C_{𝔱}^{∗,0} ×_{(S^{1},×−2)} ℂ,

where the circle action is given, for [A, Φ] ∈ C_{𝔱}^{∗,0} and ζ ∈ ℂ, by

(2.43)  ([A, Φ], ζ) ↦ ([A, e^{iθ}Φ], e^{2iθ}ζ).

Then we define the second kind of cohomology class on M_{𝔱}^{∗,0}/S^{1} by

(2.44)  μ_{c} = c_{1}(𝕃_{𝔱}) ∈ H^{2}(C_{𝔱}^{∗,0}/S^{1}; ℝ).

For monomials z ∈ 𝔸(X), we constructed [16, §3.2] geometric representatives V(z) dual to μ_{p}(z) (following the discussion in [20]) and W dual to μ_{c}, defined on M_{𝔱}^{∗}/S^{1} and M_{𝔱}^{∗,0}/S^{1}, respectively; their closures in M̄_{𝔱}/S^{1} are denoted by V̄(z) and W̄ [16, Definition 3.14]. When

(2.45)  deg(z) + 2η = dim(M_{𝔱}^{∗,0}/S^{1}) − 1,

and deg(z) ≥ dim M_{κ}^{w} it follows that [16, §3.3] the intersection

(2.46)  V̄(z) ∩ W̄^{η} ∩ M̄_{𝔱}^{∗,0}/S^{1},

is an oriented one-manifold (not necessarily connected) whose closure in M̄_{𝔱}/S^{1} can only intersect (M̄_{𝔱} − M_{𝔱})/S^{1} at points in M_{𝔱}^{red} ≅ ∪(M_{𝔰} × Sym^{ℓ}(X)), where the union is over ℓ ≥ 0 and 𝔰 ∈ Spin^{c}(X) [16, Corollary 3.18].

2.5. Donaldson invariants. We first recall the definition [20, §2] of the Donaldson series when b_{1}(X) = 0 and b_{2}^{+}(X) > 1 is odd, so that, in this case, χ + σ ≡ 0 (mod 4). See also §3.4.2 in [16], especially for a definition of the Donaldson invariants when b_{2}^{+}(X) = 1. For any choice of w ∈ H^{2}(X; ℤ), the Donaldson invariant is a linear function

D_{X}^{w} : 𝔸(X) → ℝ.

Let X̃ = X#\overline{ℂℙ}^{2} be the blow-up of X and let e ∈ H_{2}(X̃; ℤ) be the exceptional class, with Poincaré dual PD[e] ∈ H^{2}(X̃; ℤ). If z ∈ 𝔸(X) is a monomial, we define D_{X}^{w}(z) = 0 unless

(2.47)  deg(z) ≡ −2w^{2} − \frac{3}{2}(χ + σ) (mod 8).

If deg(z) obeys equation (2.47), we let κ ∈ \frac{1}{4}ℤ be defined by

deg(z) = 8κ − \frac{3}{2} (χ + σ) .

There exists an SO(3) bundle over X̃ with first Pontrjagin number −4κ − 1 and second Stiefel-Whitney class w + PD[e] (mod 2). One then defines the Donaldson invariant on monomials by

(2.48)  D_{X}^{w}(z) = # (V̄(ze) ∩ M̄_{κ+1/4}^{w+PD[e]}(X̃)) ,

and extends to a real linear function on 𝔸(X). Note that w + PD[e] (mod 2) is good in the sense of Definition 2.3. If w′ ≡ w (mod 2), then [2]

(2.49)  D_{X}^{w′} = (−1)^{\frac{1}{4} (w′−w)^{2}}D_{X}^{w}.

The Donaldson series is a formal power series,

(2.50)  D_{X}^{w}(h) = D_{X}^{w}((1 + \frac{1}{2}x)e^{h}), h ∈ H_{2}(X; ℝ).

By equation (2.47), the series D_{X}^{w} is even if

−w^{2} − \frac{3}{4}(χ + σ) ≡ 0 (mod 2),

and odd otherwise. A four-manifold has KM-simple type if for some w and all z ∈ 𝔸(X),

D_{X}^{w}(x^{2}z) = 4D_{X}^{w}(z).

According to [20, Theorem 1.7], when X has KM-simple type the series D_{X}^{w}(h) is an analytic function of h and there are finitely many characteristic cohomology classes K_{1}, …, K_{m} (the KM-basic classes) and constants a_{1}, …, a_{m} (independent of w) so that

D_{X}^{w}(h) = e^{\frac{1}{2} h·h} ∑_{i=1}^{r}(−1)^{\frac{1}{2} (w^{2}+K_{i}·w)}a_{i}e^{⟨K_{i},h⟩}.

Witten’s conjecture, [28], then relates the Donaldson and Seiberg-Witten series for fourmanifolds of simple type.

When b_{2}^{+}(X) = 1 the pairing on the right-hand side of definition (2.48) depends on the chamber in the positive cone of H^{2}(X̃; ℝ) determined by the period point of the Riemannian metric on X̃, just as in the case of Seiberg-Witten invariants described in §2.3.2. We refer to §3.4.2 in [16] for a detailed discussion of this case and, as in [16], we assume that the class w (mod 2) is good in order to avoid technical difficulties involved in chamber specification.

2.6. Links and the cobordism. Since the endpoints of the components of the onemanifold (2.46) either lie in M_{κ}^{w} or M_{𝔰} × Sym^{ℓ}(X), for some 𝔰 and ℓ(𝔱, 𝔰) ≥ 0 for which 𝔱_{ℓ} = 𝔰 ⊕ 𝔰 ⊗ L, we have when w_{2}(𝔱) is good in the sense of Definition 2.3 that

(2.51)  # (V̄(z) ∩ W̄^{n_{a}−1} ∩ L̄_{𝔱,κ}^{w}) = − ∑_{𝔰∈Spin^{c}(X)}# (V̄(z) ∩ W̄^{n_{a}−1} ∩ L̄_{𝔱,𝔰}) ,

where L̄_{𝔱,κ}^{w} is the link of M̄_{κ}^{w} in M̄_{𝔱}/S^{1} (see [15, Definition 3.7]) and where L̄_{𝔱,𝔰} is empty if ℓ(𝔱, 𝔰) < 0 and is the boundary of an open neighborhood of the Seiberg-Witten stratum M_{𝔰}×Sym^{ℓ}(X) in M̄_{𝔱}/S^{1} if ℓ = ℓ(𝔱, 𝔰) ≥ 0. By construction, the intersection of L̄_{𝔱,𝔰} with the top stratum of M̄_{𝔱}/S^{1} is a smooth manifold and the intersection of L̄_{𝔱,𝔰} with the geometric representatives is in this top stratum. The precise definition of L̄_{𝔱,𝔰} is given in [15, Definition 3.22] for ℓ = 0 and in Definition 10.2 for ℓ ≥ 1.

When deg(z) = dim M_{κ}^{w}, the intersection of the one-manifold (2.46) with the link L̄_{𝔱,κ}^{w} is given by [16, Lemma 3.30]

(2.52)  2^{1−n_{a}}# (V̄(z) ∩ W̄^{n_{a}−1} ∩ L̄_{𝔱,κ}^{w}) = # (V̄(z) ∩ M̄_{κ}^{w}) .

Applying this identity to the blow-up, X#\overline{ℂℙ}^{2}, when n_{a}(𝔱) > 0, we recover the Donaldson invariant D_{X}^{w}(z) on the right-hand side of (2.52) via definition (2.48).

3. Diagonals

In this section, we describe the stratification of the symmetric product Sym^{ℓ}(X) and the normal bundles of the strata.

3.1. Definitions. In this section, we introduce some vocabulary for describing the strata of the symmetric product Sym^{ℓ}(X).

3.1.1. Subgroups of the symmetric group. Let 𝔖_{ℓ} denote the symmetric group on ℓ elements which acts on X^{ℓ} by permutation. We will write partitions of N_{ℓ} = {1, …, ℓ} as P = {P_{1}, …, P_{r(P)}} where P_{i} ⊂ N_{ℓ}. Each such partition of N_{ℓ} gives a partition of ℓ: ℓ = |P_{1}| + … |P_{r(P)}| which we denote by |P|. The symmetric group 𝔖_{ℓ} acts on the set of partitions of N_{ℓ}, by:

(3.1)  (σ, P) → σ(P) := {σ(P_{1}), …, σ(P_{r(P)})}.

Let 𝔖(P) < 𝔖_{ℓ} be the stabilizer of a partition P with respect to the action (3.1). Note that 𝔖(P) does not preserve the ordering of the sets P_{1}, …, P_{r(P)}. In addition, let Γ(P) be the subgroup of 𝔖_{ℓ} defined by the image of the inclusion:

(3.2)  ∏_{P ∈P} 𝔖_{|P |} → 𝔖_{ℓ},

given by 𝔖_{|P |} acting trivially on the elements of N_{ℓ} not in P. Lemma 3.1. Let P be a partition of N_{ℓ}. Then the subgroup Γ(P) of 𝔖_{ℓ} defined by the image of the inclusion (3.2) is the stabilizer of P under the action (3.1).

Proof. Let Γ′(P) be the stabilizer of P under the action (3.1). There is then an inclusion Γ(P) → Γ′(P). This map is surjective because any σ ∈ 𝔖_{ℓ} which preserves the subsets P_{1}, …, P_{r(P)} is a permutation of each of these subsets and thus in Γ(P). □

Lemma 3.2. Let P = {P_{1}, …, P_{r(P)}} be a partition of N_{ℓ}. The action (3.1) defines an action of Γ(P) on the set {P_{1}, …, P_{r(P)}}. Let 𝔖(|P|) be the subgroup of 𝔖_{r(P)} which preserves the partition |P_{1}|, …, |P_{r(P)}|. Then Γ(P) is a normal subgroup of 𝔖(P) and there is an isomorphism:

𝔖(|P|) ≃ 𝔖(P)/Γ(P).

Proof. Let t : 𝔖(P) → 𝔖_{r(P)} be the homomorphism defined by the action of 𝔖(P) on the set {P_{1}, …, P_{r(P)}}. The kernel of t is the subgroup of elements preserving the sets P_{1}, …, P_{r(P)} which is Γ(P) by Lemma 3.1, implying Γ(P) is a normal subgroup of 𝔖(P). Because each σ ∈ 𝔖(P) is a bijection of N_{ℓ}, t(σ) preserves the partition |P|. Thus, t(𝔖(P)) ⊂ 𝔖(|P|). To prove surjectivity, given σ ∈ 𝔖(|P|), construct σ̃ ∈ 𝔖(P) with t(σ̃) = σ as follows. Define an order relation on {P_{1}, …, P_{r(P)}} by comparing the smallest elements (with respect to the natural order on N_{ℓ}) of elements of P_{i} and P_{j}. If a ∈ N_{ℓ} is the k-th element of P_{i} and σ(i) = j, define σ̃(a) to be the k-th element of P_{j}. One then checks σ̃ ∈ 𝔖(P) and t(σ̃) = σ. □

3.1.2. Defining the diagonals. The strata of Sym^{ℓ}(X) are defined by the quotients of diagonals in X^{ℓ} by the action of 𝔖_{ℓ} on X^{ℓ}. Explicitly, define: (3.3)

Δ^{∘}(X^{ℓ}, P) = {(x_{1}, …, x_{ℓ}) ∈ X^{ℓ} : x_{i} = x_{j} if and only if there is P ∈ P with i, j ∈ P },
Δ(X^{ℓ}, P) = {(x_{1}, …, x_{ℓ}) ∈ X^{ℓ} : x_{i} = x_{j} if there is P ∈ P with i, j ∈ P }.

The preceding subspaces are related by

(3.4)  cl_{X^{ℓ}}Δ^{∘}(X^{ℓ}, P) = Δ(X^{ℓ}, P).

We can identify Δ^{∘}(X^{ℓ}, P) with the complement of the big diagonal in X^{r(P)} as follows. Write P as an ordered collection P_{1}, …, P_{r(P)}. Then define an embedding,

(3.5)  ι_{P} : X^{r} \ ∪_{i<j} {x_{i} = x_{j}} → Δ^{∘}(X^{ℓ}, P) ⊂ X^{ℓ},

by ι(x_{1}, …, x_{r}) = (y_{1}, …, y_{ℓ}) where y_{k} = x_{i} for k ∈ P_{i}. The map ι_{P} is a diffeomorphism. Lemma 3.3. Let P be a partition of N_{ℓ}. Then:

(1) The diagonal Δ(X^{ℓ}, P) is the fixed point set of the group Γ(P),

(2) If x ∈ Δ^{∘}(X^{ℓ}, P), then Stab_{x} = Γ(P),

(3) If σ ∈ 𝔖_{ℓ} then σ(Δ^{∘}(X^{ℓ}, P)) ⊂ Δ^{∘}(X^{ℓ}, P) if and only if σ ∈ 𝔖(P).

Proof. The fixed point set of 𝔖_{i} acting on X^{i} is the small diagonal. Thus, by the definition of Γ(P), if x = (x_{1}, …, x_{ℓ}) is a fixed point for every σ ∈ Γ(P), we have x_{i} = x_{j} for all i, j ∈ P for every P ∈ P, implying the first assertion.

The first assertion implies Γ(P) ⊂ Stab_{x} for x ∈ Δ^{∘}(X^{ℓ}, P). If σ ∈ Stab_{x} and σ ∉ Γ(P), then there is P ∈ P and i ∈ P such that σ(i) ∉ P. But σ(x) = x then implies x_{σ(i)} = x_{i} which contradicts x ∈ Δ^{∘}(X^{ℓ}, P). This implies the second assertion.

By definition, x ∈ Δ^{∘}(X^{ℓ}, P) if and only if σ(x) ∈ Δ^{∘}(X, σ(P)). □

3.1.3. Strata of the symmetric product. Let π_{𝔖} : X^{ℓ} → Sym^{ℓ}(X) = X^{ℓ}/𝔖_{ℓ} be the projection. Define:

(3.6)  Σ(X^{ℓ}, P) = π_{𝔖} (Δ^{∘}(X^{ℓ}, P)) .

From item (3) of Lemma 3.3, we have the relation:

(3.7)  Σ(X^{ℓ}, P) = Δ^{∘}(X^{ℓ}, P)/𝔖(P).

The action of 𝔖(P) on Δ^{∘}(X^{ℓ}, P) is not free as noted in item (2) of Lemma 3.3. From Lemma 3.2, 𝔖(|P|) is the natural group acting on Δ^{∘}(X^{ℓ}, P). We see this action as follows. Lemma 3.4. Choose an ordering P_{1}, …, P_{r(P)} of the partition P and use the ordering |P_{1}|, …, |P_{r(P)}| of |P|. Then the map ι_{P} given in (3.5) is equivariant with respect to the actions of 𝔖(|P|) on X^{r} and 𝔖(P) on Δ^{∘}(X^{ℓ}, P).

The proof of the following relations between Σ(X^{ℓ}, P) and Δ^{∘}(X^{ℓ}, P) is elementary. Lemma 3.5. Let P be a partition of N_{ℓ} and let [P] denote the orbit of P under 𝔖_{ℓ}. Then,

(1) If P′ = σ(P), then Γ(P) = σΓ(P′)σ^{−1},

(2) If P′ = σ(P), then 𝔖(P) = σ𝔖(P′)σ^{−1},

(3) π_{𝔖}^{−1} (Σ(X^{ℓ}, P)) = ∪_{P′∈[P]}Δ^{∘}(X, P′).

3.2. Incidence relations. We now describe incidence relations among the diagonals of X^{ℓ} and the resulting relations among the strata Σ of Sym^{ℓ}(X). Definition 3.6. Let P and P′ be partitions of N_{ℓ}. We say P′ is a refinement of P, or P < P′, if for every P′ ∈ P′ there is P ∈ P with P′ ⊆ P. Lemma 3.7. If P and P′ are partitions of N_{ℓ} then Δ^{∘}(X^{ℓ}, P) ⊂ Δ(X, P′) if and only P′ is a refinement of P.

Proof. If P′ is a refinement of P and x ∈ Δ^{∘}(X^{ℓ}, P), then for every P′ ∈ P and i, j ∈ P′, there is P ∈ P with i, j ∈ P′ ⊂ P. Since i, j ∈ P ∈ P, we have x_{i} = x_{j} so x ∈ Δ(X, P′) proving one implication.

Conversely, if Δ^{∘}(X^{ℓ}, P) ⊂ Δ(X, P′) and P′ ∈ P′, then because every x = (x_{1}, …, x_{ℓ}) ∈ Δ^{∘}(X^{ℓ}, P) is also in Δ(X, P′) we see that for every i, j ∈ P′, x_{i} = x_{j}. This implies (by the only if in the definition of Δ^{∘}(X^{ℓ}, P) that there must be P ∈ P with i, j ∈ P. Thus, P′ ⊂ P and P < P. □

Now, we compare the incidence relations with the strata of Sym^{ℓ}(X). Recall that for a partition P of N_{ℓ}, [P] denoted the orbit of P under the action (3.1) of 𝔖_{ℓ} on the set of partitions of N_{ℓ}. For P and P′ partitions of N_{ℓ}, define

(3.8)  [P < P′] = {P′′ ∈ [P′] : P < P′′}.

Remark 3.8. Observe that 𝔖(P) acts transitively on [P < P′] by the action (3.1).

The next lemma follows from the definition of Σ(X^{ℓ}, P) (see (3.6)) as well as Lemma 3.7 and item (3) in Lemma 3.5.

Lemma 3.9. If P and P′ are partitions of N_{ℓ} then

Σ(X^{ℓ}, P) ⊂ cl_{Sym^{ℓ}(X)}Σ(P′)

if and only if there is P′′ ∈ [P] with P′ < P′′.

3.3. Normal bundles. We now introduce the normal bundles of the strata Δ^{∘}(X^{ℓ}, P) of X^{ℓ}. Lemma 3.10. Let P be any partition of N_{ℓ}. The tangent bundle of the submanifold Δ^{∘}(X^{ℓ}, P) is given by: (3.9) T Δ^{∘}(X^{ℓ}, P) = {(v_{1}, …, v_{ℓ}) ∈ T X^{ℓ} : v_{i} = v_{j} if and only if there is P ∈ P with i, j ∈ P }.

If g is any metric on X, then the orthogonal complement of T Δ^{∘}(X^{ℓ}, P) with respect to the metric on X^{ℓ} given by the ℓ-fold product of g is the following normal bundle of Δ^{∘}(X^{ℓ}, P):

(3.10)  ν̃(X^{ℓ}, P) := {(v_{1}, …, v_{ℓ}) ∈ T X^{ℓ}|_{Δ^{∘}(X^{ℓ},P)} : ∑_{i∈P} v_{i} = 0 for all P ∈ P}.

Proof. Differentiate any path in Δ^{∘}(X^{ℓ}, P) to see the form of T Δ^{∘}(X^{ℓ}, P) appearing in equation (3.9). For v⃗ = (v_{1}, …, v_{ℓ}) ∈ T Δ^{∘}(X^{ℓ}, P), we then write v_{P} for any v_{i} with i ∈ P. Then, if v⃗ = (v_{1}, …, v_{ℓ}) ∈ T Δ^{∘}(X^{ℓ}, P) and w⃗ ∈ ν̃(X^{ℓ}, P), the equality

v⃗ · w⃗ = ∑_{i} v_{i} · w_{i} = ∑_{P ∈P} v_{P} · (∑_{i∈P} w_{i}) = 0,

which holds for any product metric on X^{ℓ}, implies that the bundle in equation (3.10) is contained in the orthogonal complement of T Δ^{∘}(X^{ℓ}, P). Since the above inclusion must hold for any value of v_{P}, we see that any element of the orthogonal complement of T Δ^{∘}(X^{ℓ}, P) must satisfy the equations defining the bundle in (3.10). □

There is an obvious 𝔖(P) action on the normal bundle ν̃(X^{ℓ}, P). We will write ν(X^{ℓ}, P) for the quotient ν̃(X^{ℓ}, P)/𝔖(P). Lemma 3.11. There is an open neighborhood, Õ(X^{ℓ}, P) ⊂ ν̃(X^{ℓ}, P), of the zero-section and an 𝔖(P)-equivariant exponential map identifying Õ(X^{ℓ}, P) with an open neighborhood, Ũ(X^{ℓ}, P), of Δ^{∘}(X^{ℓ}, P) in X^{ℓ}.

Because the exponential map is 𝔖(P) equivariant, equation (3.7) implies: Lemma 3.12. Continue the notation of Lemma 3.11. If O(X^{ℓ}, P) = Õ(X^{ℓ}, P)/𝔖(P), then a neighborhood,

U(X^{ℓ}, P) = Ũ(X^{ℓ}, P)

of Σ(X^{ℓ}, P) in Sym^{ℓ}(X) is homeomorphic to O(X^{ℓ}, P).

We now identify the intersection of Δ^{∘}(X, P′) with Ũ(X^{ℓ}, P). Lemma 3.13. If P < P′ are partitions of N_{ℓ} and (3.11)

ν̃(X^{ℓ}, P, P′)

= {(v_{1}, …, v_{ℓ}) ∈ ν̃(X^{ℓ}, P) : v_{i} = v_{j} if and only if there is P′ ∈ P′ with i, j ∈ P′}.

then the intersection of Δ^{∘}(P′) with Ũ(X^{ℓ}, P) is homeomorphic to ν̃(X^{ℓ}, P, P′) ∩ Õ(X^{ℓ}, P).

Proof. This follows immediately from the equivariance of the exponential map. □

Thus, a neighborhood of Δ^{∘}(X^{ℓ}, P′) near the lower diagonal Δ^{∘}(X^{ℓ}, P) can be described by the bundle ν̃(X^{ℓ}, P, P′). However, the end of the stratum Σ(X^{ℓ}, P′) near the lower stratum Σ(X^{ℓ}, P) can be more complicated than a quotient of ν̃(X^{ℓ}, P, P′) by a subgroup of 𝔖_{ℓ}. Recall that the pre-image of Σ(X^{ℓ}, P) is not just Δ^{∘}(X^{ℓ}, P) but rather all the diagonals given by a partition in [P], the orbit of P under the action (3.1). Thus, while we can cover Σ(X^{ℓ}, P) by Δ^{∘}(X^{ℓ}, P), to account for all the ends of Δ^{∘}(X^{ℓ}, P′) which get mapped to a neighborhood of the lower stratum, Σ(X^{ℓ}, P), we must consider all diagonals Δ^{∘}(X^{ℓ}, P′) where P′′ ∈ [P′] and P < P′. That is, we must consider all partitions P′′ ∈ [P < P′], where [P < P′] is defined in (3.8). The next lemma follows from Lemmas 3.11 and 3.13. Lemma 3.14. Let P < P′ be partitions of N_{ℓ}. Let [P < P′] be the set of partitions defined in (3.8). Then a neighborhood of Σ(X^{ℓ}, P) in Σ(X^{ℓ}, P′) is homeomorphic to a neighborhood of the zero-section in

(3.12)  ν(X^{ℓ}, P, [P′]) = (⊔_{P′′∈[P<P′]}ν̃(X^{ℓ}, P, P′)) /𝔖(P).

4. The Thom-Mather structure of the symmetric product

In this section, we define a partial Thom-Mather structure on the symmetric product. In general, by a partial Thom-Mather structure on a stratified space Z with strata Σ_{i}, we mean neighborhoods U_{i} of Σ_{i} with tubular neighborhood projections π_{i} : U_{i} → Σ_{i} and tubular distance functions t_{i} : U_{i} → [0, 1) satisfying t_{i}^{−1}(0) = Σ_{i} and for Σ_{i} ⊂ cl_{Z}(Σ_{j}) the compatability conditions

(4.1)  \begin{aligned}π_{i} ∘ π_{j} & = π_{i}, \\ t_{i} ∘ π_{j} & = t_{i}. \end{aligned}

If t and t_{r} are tubular distance functions of a tubular neighborhood π : U → Σ, then we say t_{r} is a rescaling of t if there is a non-vanishing function f : Σ → (0, 1] such that

(4.2)  t_{r}(·) = t(·)f(π(·)).

Note that an actual Thom-Mather structure has a number of additional requirements on the derivatives of the maps π_{i} and t_{i}. While it is well-known that Sym^{ℓ}(X), being the quotient of the smooth, compact manifold X^{ℓ} by a finite group 𝔖_{ℓ}, is a Thom-Mather stratified space, the precise construction we present here will be useful in following sections.

We begin by defining subspaces of Sym^{κ}(ℝ^{4}) which will appear as the fibers of the normal bundles of §3.3. We construct partial Thom-Mather structures on this subspace in §4.1. To use the normal bundles of §3.3 to construct Thom-Mather structures on Sym^{ℓ}(X), we first introduce some deformations of the metric in §4.2. We describe the overlap of the exponential maps from these normal bundles in §4.3. Finally, we produce the partial Thom-Mather structure on Sym^{ℓ}(X) in §4.4.

4.1. Diagonals in ℝ^{4}. The fiber of the normal bundles in §3.3 are defined by the subspace of mass-centered points in Sym^{κ}(ℝ^{4}). Hence, for any subset P of N_{ℓ}, define

z_{P} : ⊕_{i∈P} ℝ^{4} → ℝ^{4}, z_{P} ((v_{i})_{i∈P}) = ∑_{i} v_{i}.

and

(4.3)  Z_{P} = z_{P}^{−1}(0) ⊂ ⊕_{i∈P} ℝ^{4}.

We will write z_{ℓ} for z_{N_{ℓ}}. We define the cone point

(4.4)  c_{P} ∈ Z_{P}

to be the zero vector in ⊕_{i∈P} ℝ^{4}. The fiber of ν̃(X^{ℓ}, P) is then given by (compare (3.10))

(4.5)  ∏_{P ∈P} Z_{P} .

For P < P′ partitions of N_{ℓ}, define a partition of each P ∈ P by

(4.6)  P′_{P} = {P′ ∈ P′ : P′ ⊆ P }.

If P ∈ P ∩ P′, then P′_{P} = {P }. If we define (4.7)

Δ^{∘}(Z_{P} , P′_{P}) = {(v_{i})_{i∈P} ∈ Z_{P} : v_{i} = v_{j} if and only if there is P′ ∈ P′_{P} with i, j ∈ P′},

then the fiber of the bundle ν̃(X^{ℓ}, P, P′) of equation (3.11) is given by:

(4.8)  ∏_{P ∈P} Δ^{∘}(Z_{P} , P′_{P}).

Note that if P′_{P} is given by one set, P, as is the case if P ∈ P ∩ P′, then Δ^{∘}(Z_{P} , P′_{P}) is a single point, given by |P | copies of the zero vector in ℝ^{4}. The following lemma describes the normal bundle of Δ^{∘}(Z_{P} , P′_{P}) ⊂ z_{P}^{−1}(0). Lemma 4.1. Let P′_{P} be a partition of P. Then the restriction of the map

(4.9)  e(Z_{P} , P′_{P}) : Δ^{∘}(Z_{P} , P′_{P}) × ∏_{P′∈P′_{P}} Z_{P′} → Z_{P}

defined by

(4.10)  ((v_{i})_{i∈P} , (z_{j})_{j∈P′,P′∈P′_{P}} ) → (v_{i} + z_{i})_{i∈P}

to a neighborhood O(Z_{P} , P′_{P}) of the subspace

Δ^{∘}(Z_{P} , P′_{P}) × ∏_{P′∈P′_{P}} {c_{P′}}

is a homeomorphism onto its image.

Proof. We observe the map (4.10) can be inverted by the center of mass map

(4.11)  \begin{aligned}(w_{i})_{i∈P} → & ((v_{i})_{i∈P} , (z_{i})_{i∈P′,P′∈P′_{P}} ) \\ v_{i} & = \frac{1}{|P′|} ∑_{j∈P′} w_{j}, for i ∈ P′, \\ z_{j} & = w_{j} − v_{j}. \end{aligned}

(Note that the above map is an inverse because ∑_{j∈P′} z_{j} = 0 by the definition of Z_{P′}.) However, the map (4.11) only takes values in

Δ(Z_{P} , P′_{P}) × ∏_{P′∈P′_{P}} Z_{P′}

unless we restrict to a sufficiently small neighborhood O(Z_{P} , P′_{P}). □

Because Z_{P} is a normal topological space, we can find open neighborhoods U(Z_{P} , P′_{P}) of Δ^{∘}(Z_{P} , P′_{P}) for every partition P′_{P} of P such that U(Z_{P} , P′_{P}) ∩ U(Z_{P} , P′′_{P}) is empty unless P′_{P} < P′′_{P} or P′′_{P} < P′_{P}. The normal bundle structure given by the map (4.10) then defines a tubular neighborhood projection,

(4.12)  π(Z_{P} , P′_{P}) : U(Z_{P} , P′_{P}) → Δ^{∘}(Z_{P} , P′_{P}),

given by the v_{i} component of the map (4.11):

(4.13)

π(Z_{P} , P′_{P}) ((w_{i})_{i∈P}) = (v_{i})_{i∈P} ,

where for i ∈ P′ ∈ P′_{P}, v_{i} = \frac{1}{|P′|} ∑_{j∈P′} w_{j}.

The following lemma shows that the maps π(Z_{P} , P′_{P}) satisfy the first equality of (4.1). Lemma 4.2. If P′_{P} < P′′_{P} are partitions of P, then on the intersection U(Z_{P} , P′_{P}) ∩ U(Z_{P} , P′′_{P}), the equality

(4.14)  π(Z_{P} , P′_{P}) ∘ π(Z_{P} , P′′_{P}) = π(Z_{P} , P′_{P})

holds.

Proof. The lemma is immediate from the definition (4.13). □

We use the above set of consistent tubular neighborhoods to define a scale function which is constant on the fibers of the maps π(Z_{P} , P′_{P}). Lemma 4.3. There is a smooth function t(Z_{P}) : Z_{P} → [0, ∞) and open neighborhoods U′(Z_{P} , P′_{P}) ⋐ U(Z_{p}, P′_{P}) of the diagonals Δ^{∘}(Z_{P} , P′_{P}) satisfying

(1) If c_{P} ∈ Z_{P} is the cone point defined in (4.4), t(Z_{P})^{−1}(0) = c_{P},

(2) For every partition P′_{P} of P, t(Z_{P}) ∘ π(Z_{P} , P′_{P}) = t(Z_{P}), on U′(Z_{P} , P′_{P})

(3) The function is invariant with respect to the permutation action of 𝔖_{|P |} and the diagonal rotation action of SO(4) on Z_{P}.

Proof. By induction on |P |, we can assume the functions t(Z_{P′}) are already defined if |P′| < |P |. (For |P | = 1, Z_{P} = {c_{P} } and t(Z_{P}) must then be the zero function.) If

π_{P′} : Δ^{∘}(Z_{P} , P′_{P}) × ∏_{P′∈P′_{P}} Z_{P′} → Z_{P′}

is the projection map, then the map t(Z_{P} , P′_{P}) : U(Z_{P} , P′_{P}) → [0, ∞) defined by

t(Z_{P} , P′_{P}) = ( ∑_{P′∈P′_{P}} |P′|(t(Z_{P′}) ∘ π_{P′})^{2}) ∘ e(Z_{P} , P′_{P})^{−1},

where e(Z_{P} , P′_{P}) is defined in (4.9), is a tubular distance function. By rescaling t(Z_{P} , P′_{P}) we can obtain a tubular distance function t_{r}(Z_{P} , P′_{P}) : U(Z_{P} , P′_{p}) → [0, ∞) satisfying t_{r}(Z_{P} , P′_{P})^{−1}(0) = Δ^{∘}(Z_{P} , P′_{P}), and t_{r}(Z_{P} , P′_{P})^{−1}([0, \frac{1}{2})) = U′(Z_{P} , P′_{P}) ⋐ U(Z_{P} , P′_{P}).

Enumerate the partitions P′_{P} of P as P_{1}, …, P_{r} such that if P_{i} < P_{j}, then i < j. Let β : [0, 1] → [0, 1] be a smooth function with β([0, \frac{1}{4}]) = 1 and β([\frac{1}{2}, 1]) = 0. If we define β_{i} = β ∘ t_{r}(Z_{P} , P′_{i}), then U′(Z_{P} , P_{i}) ⊂ β_{i}^{−1}(1), and β_{i} is compactly supported on U(Z_{P} , P_{i}).

We construct functions t_{i} : Z_{P} → [0, ∞) using upwards induction on i. Begin with the standard scale, t_{0}((v_{i})_{i∈P}) = ∑_{i} |v_{i}|^{2}. Assume t_{i−1} satisfies

(4.15)  t_{i−1} ∘ π(Z_{P} , P_{j}) = t_{i−1} for all j < i

and the invariance conditions in the conclusion of the lemma. We construct t_{i} by

(4.16)  t_{i}(·) = β_{i}(·)t_{i−1} ∘ π(Z_{P} , P_{i})(·) + (1 − β_{i}(·)) t_{i−1}(·).

Observe that by the inclusion U′(Z_{P} , P_{i}) ⊂ β_{i}^{−1}(1), t_{i} = t_{i−1} ∘ π(Z_{P} , P_{i}) on U′(Z, P_{i}) and thus, because π(Z_{P} , P_{i}) ∘ π(Z_{P} , P_{i}) = π(Z_{P} , P_{i}), t_{i} ∘ π(Z_{P} , P_{i}) = t_{i} on U′(Z, P_{i}) on U′(Z_{P} , P_{i}). By construction, t_{i} is SO(4)-invariant. We make t_{i} 𝔖(P)-invariant by defining all t_{i′} for P_{i′} conjugate to P_{i} by the obvious conjugation. Finally, we observe that for j < i, the equality

π(Z_{P} , P_{j}) ∘ π(Z_{P} , P_{i}) = π(Z_{P} , P_{j})

(see (4.14)) and the inductive assumption (4.15) imply that on U′(Z_{P} , P_{j}),

t_{i−1} ∘ π(Z_{P} , P_{i}) = t_{i−1} ∘ π(Z_{P} , P_{j}) ∘ π(Z_{P} , P_{i}) = t_{i−1} ∘ π(Z_{P} , P_{j}) = t_{i−1}.

The interpolation between t_{i−1} and t_{i−1} ∘ π(Z_{P} , P_{i}) in (4.16) thus leaves t_{i} equal to t_{i−1} on U′(Z_{P} , P′_{j}). Hence, for j < i,

t_{i} ∘ π(Z_{P} , P_{j})|_{U′(Z_{P} ,P_{j})} = t_{i−1}π(Z_{P} , P_{j})|_{U′(Z_{P} ,P_{j})} = t_{i−1}|_{U′(Z_{P} ,P_{j})},

completing the induction. □

4.2. Families of metrics. A metric g on X determines a product metric on X^{ℓ} and thus exponential maps of the normal bundles of the diagonal. We will use a non-standard diffeomorphism of the normal bundle ν̃(X^{ℓ}, P) with a neighborhood of Δ^{∘}(X^{ℓ}, P) in X^{ℓ}. This diffeomorphism will be defined by varying the metric on X and thus the resulting product metric on X^{ℓ}. We begin by noting that it is possible to define an exponential map with a varying metric. Lemma 4.4. If g_{P} is a smooth family of metrics on X parameterized by x ∈ Δ^{∘}(X^{ℓ}, P), then there is an open neighborhood Õ(X^{ℓ}, P, g_{P}) ⊂ ν̃(X^{ℓ}, P) of the zero-section and a smooth embedding,

(4.17)  e(X^{ℓ}, g_{P}) : Õ(X^{ℓ}, P, g_{P}) ⊂ ν̃(X^{ℓ}, P) → X^{ℓ},

parameterizing an open neighborhood, Ũ(X^{ℓ}, P, g_{P}), of Δ^{∘}(X^{ℓ}, P) in X^{ℓ} such that the restriction of the map e(X^{ℓ}, g_{P}) to a fiber Õ(X^{ℓ}, P, g_{P})|_{x} is given by the exponential map defined by the ℓ-fold product of the metric g_{P,x}.

Proof. For any metric g′ on X, the derivative of the exponential map at a point in the zerosection is the identity. Thus, the derivative of this exponential map with varying product metric is also the identity and this map defines an embedding of a neighborhood of the zero-section. □

The image of the exponential map in (4.17) is a tubular neighborhood. Such a tubular neighborhood, Ũ(X^{ℓ}, P, g_{P}) ⊂ X^{ℓ}, defines a projection π(X^{ℓ}, g_{P}) : Ũ(X^{ℓ}, P, g_{P}) → Δ^{∘}(X^{ℓ}, P) by the composition of e(X^{ℓ}, g_{P})^{−1} and the projection ν̃(X^{ℓ}, P) → Δ^{0}(X, P).

For any smooth n-manifold M, let Fr_{GL}(T M) denote the GL(n)-bundle of frames of tangent vectors. Define Fr_{Gl}(T X^{ℓ}, P) as:

Fr_{GL}(T X^{ℓ}, P) = {(F_{1}, …, F_{ℓ}) ∈ Fr(T X^{ℓ})|_{Δ^{∘}(X^{ℓ},P)} : F_{i} = F_{j} for all i, j ∈ P ∈ P}.

If g_{P} is a family of metrics parameterized by Δ^{∘}(X^{ℓ}, P), then we define Fr(T X^{ℓ}, P, g_{P}) to be the subbundle of Fr_{GL}(T X^{ℓ}, P) of frames such that if (F_{1}, …, F_{ℓ}) ∈ Fr(T X^{ℓ}, P, g_{P})|_{x} then F_{i} is orthonormal with respect to g_{P,x}. The structure group of Fr(T X^{ℓ}, P, g_{P}) is

G(T, P) = SO(4)^{r(P)}, where P = {P_{1}, …, P_{r(P)}}.

Then we have the identity:

(4.18)  ν̃(X^{ℓ}, P) = Fr(T X^{ℓ}, P, g_{P}) ×_{G(T,P)} ∏_{P ∈P} Z_{P} .

Then Lemma 3.13 implies the following.

Lemma 4.5. Let P < P′ be partitions of N_{ℓ}. Then, the restriction of the exponential map e(X^{ℓ}, g_{P}) to

(4.19)  Õ(X^{ℓ}, P, g_{P}) ∩ (Fr(T X^{ℓ}, P, g_{P}) ×_{G(T,P)} ∏_{P ∈P} Δ^{∘}(Z_{P} , P′_{P})) ,

takes values in Δ(X, P′).

Define:

G(T, P′_{P}) = ∏_{P′∈P′_{P}} SO(4).

Then, we have:

Lemma 4.6. Let P < P′ be partitions of N_{ℓ}. Let Ũ(X, P, P′) ⊂ Δ^{∘}(X^{ℓ}, P′) be the image of the restriction of the exponential map in (4.19). Let g_{P} and g_{P′} be smooth families of metrics parameterized by Δ^{∘}(X^{ℓ}, P) and Δ^{∘}(X, P′) respectively. Assume that g_{P′,x′} = g_{P,x} if π(X^{ℓ}, g_{P})(x′) = x. Then,

(4.20) Fr(T X^{ℓ}, P′, g_{P′})|_{Ũ(X,P,P′)} ≃ Fr(T X^{ℓ}, P, g_{P})×_{G(T,P)} ∏_{P ∈P} (Δ^{∘}(Z_{P} , P′_{P}) × G(T, P′_{P})) .

Proof. By Lemma 4.5, a neighborhood of Δ^{∘}(X^{ℓ}, P) in Δ^{∘}(X^{ℓ}, P′) is diffeomorphic to a neighborhood of the zero-section in

(4.21)  ν̃(X^{ℓ}, P, P′) ≅ Fr(T X^{ℓ}, P, g_{P}) ×_{G(T,P)} ∏_{P ∈P} Δ^{∘}(Z_{P} , P′_{P})

by the exponential map e(X^{ℓ}, g_{P}). The isomorphism (4.20) is then defined by the obvious parallel translation of the frames in Fr(T X^{ℓ}, P, g_{P}) to the point in Ũ(X^{ℓ}, P, P′) given by the data in (4.21). □

4.3. Overlap maps. Let P < P′ be partitions of N_{ℓ}. To define tubular neighborhoods and projection maps which satisfy the conditions (4.1), we define a space to control the overlap of the tubular neighborhoods U(X, P, g_{P}) and U(X, P′, g_{P′}) as follows.

(4.22) ν̃(X^{ℓ}, P, P′, g_{P}) = Fr(T X^{ℓ}, P, g_{P}) ×_{G(T,P)} ∏_{P ∈P} (Δ^{∘}(Z_{P} , P′_{P}) × ∏_{P′∈P′_{P}} Z_{P′} ) .

We define an open subspace Õ(X^{ℓ}, P, P′) of ν̃(X^{ℓ}, P, P′) and maps ρ_{P,P′}^{X,u} and ρ_{P,P′}^{X,d} so that the diagram

(4.23)  \begin{matrix}Õ(X^{ℓ}, P, P′, g_{P}) & ⟶^{ρ_{P,P′}^{X,u}} & Õ(X^{ℓ}, P′, g_{P′}) \\ ^{ρ_{P,P′}^{X,d}} ↓ & ^{e(X^{ℓ},g_{P′})}↓ \\ Õ(X^{ℓ}, P, g_{P}) & ⟶^{e(X^{ℓ},g_{P′})} & X^{ℓ} \end{matrix}

commutes and controls the overlaps of the images of the exponential maps e(X^{ℓ}, g_{P}) and e(X^{ℓ}, g_{P′}).

4.3.1. The downwards overlap map. The downwards overlap map,

(4.24)  ρ_{P,P′}^{X,d} : Õ(X^{ℓ}, P, P′, g_{P}) → Õ(X^{ℓ}, P, g_{P})

is given by the fiberwise inclusions of the map of equation (4.9),

Δ^{∘}(Z_{P} , P′_{P}) × ∏_{P′∈P′_{P}} Z_{P′} → Z_{P} .

Because this map is SO(4) equivariant, it extends to the domain given in (4.24). The open subspace Õ(X^{ℓ}, P, P′, g_{P}) must satisfy

(4.25)  Õ(X^{ℓ}, P, P′, g_{P}) ⊆ (ρ_{P,P′}^{X,d})^{−1} (Õ(X^{ℓ}, P, g_{P}))

for the composition in the diagram (4.23) to be defined.

4.3.2. The upwards overlap map. We define the upwards overlap map,

(4.26)  ρ_{P,P′}^{X,u} : ν̃(X^{ℓ}, P, P′, g_{P}) → Õ(X^{ℓ}, P, g_{P})

as follows. Let

(4.27)  π_{1} : ν̃(X^{ℓ}, P, P′) → ν̃(X^{ℓ}, P, P′)

be the projection implicit in (4.21) and (4.22). We assume the open set Õ(X^{ℓ}, P, P′, g_{P}) satisfies

(4.28)  Õ(X^{ℓ}, P, P′, g_{P}) ⊆ π_{1}^{−1} (Õ(X^{ℓ}, P → P′)) .

Then, the composition e(X^{ℓ}, g_{P}) ∘ π_{1} is defined on Õ(X^{ℓ}, P, P′, g_{P}) and takes values in Δ^{∘}(X^{ℓ}, P′).

We now define the map ρ_{P,P′}^{X,u}. We make the assumption on the families of metrics g_{P} and g_{P′} appearing in Lemma 4.6. Given a point in ν̃(X^{ℓ}, P, P′, g_{P}), lying over y = (y_{P}) ∈ Δ^{∘}(X^{ℓ}, P), define a point y′ = (y_{P′})_{P′∈P′} ∈ Δ^{∘}(X^{ℓ}, P′) by the composition

e(X^{ℓ}, g_{P}) ∘ π_{1} : ν̃(X^{ℓ}, P, P′, g_{P}) → Δ^{∘}(X^{ℓ}, P′).

Then, for P′ ∈ P′_{P} (where P ∈ P), define F_{P′} to be the parallel translate of the frame F_{P} ∈ Fr(T X)|_{y_{P}} from y_{P} to y′_{P′} using the Levi-Civita connection defined by the metric g_{P,y}. Leave the data in Z_{P′} alone. The result is an element of

Fr(T X^{ℓ}, sP′, g_{P′})|_{y′} ×_{G(T,P′)} ∏_{P′∈P′} Z_{P′} = ν̃(X^{ℓ}, P′, g_{P′})|_{y′},

thus defining the map ρ_{P,P′}^{X,u}.

4.3.3. Commuting overlaps maps. We now define conditions on the metrics g_{P} and g_{P′} which ensure that the diagram (4.23) commutes.The diagram will only commute if we chose the families of metrics g_{P} and g_{P′} in such a way as to eliminate holonomy. To do this, we introduce the notion of a locally flattened metric. Definition 4.7. If A ⊂ X^{ℓ}, then the support of A in X is the union of the images of A under the projection maps X^{ℓ} → X.

Let g_{P} be a smooth family of metrics on X parameterized by Δ^{∘}(X^{ℓ}, P). Let Ũ′ ⊆ Ũ(X^{ℓ}, P, g_{P}) be a neighborhood of Δ^{∘}(X^{ℓ}, P). The family of metrics g_{P} is locally flat with respect to Ũ′ at x ∈ Δ^{∘}(X^{ℓ}, P) if the metric g_{P,x} is flat on the support of Ũ′ ∩ π_{P}^{−1}(x) in X. The family of metrics is locally flat with respect to Ũ′ if this holds for all x ∈ Δ^{∘}(X^{ℓ}, P).

The following lemma shows that the diagram (4.23) commutes when the metrics are flat in the sense of the preceding definition. Lemma 4.8. Let P < P′ be partitions of N_{ℓ}. Assume that the smooth families of metrics g_{P} and g_{P′} satisfy

(1) The metrics g_{P} are flat with respect to a neighborhood Ũ′ ⊆ Ũ(X^{ℓ}, P, g_{P}),

(2) For y′ ∈ Ũ(X^{ℓ}, P, g_{P′}) ∩ Δ^{∘}(X^{ℓ}, P′) with π(X^{ℓ}, g_{P})(y′) = y, the metrics satisfy

g_{P,y} = g_{P′,y′}.

Then there is a neighborhood Õ′(X, P, P′, g_{P}) of the zero-section of the bundle (4.22) such that the diagram (4.23) commutes when restricted to Õ′(X, P, P′, g_{P}).

Proof. We observe that for the compositions,

e(X^{ℓ}, g_{P}) ∘ ρ_{P,P′}^{X,d} and e(X^{ℓ}, g_{P′}) ∘ ρ_{P,P′}^{X,u}

to be defined, the neighborhood Õ′(X, P, P′, g_{P}) must satisfy the inclusion relations (4.25) and (4.28). However, we replace the condition (4.25) with the stronger requirement that

(4.29)  Õ′(X, P, P′, g_{P}) ⊆ (e(X^{ℓ}, g_{P}) ∘ ρ_{P,P′}^{X,d})^{−1}(Ũ′).

On any open subset satisfying the above condition, the composition

e(X^{ℓ}, g_{P}) ∘ ρ_{P,P′}^{X,d}

is defined by the exponential map of the metric g_{P,y} of the vectors obtained by the appropriate adding of the vectors in the fiber of ν̃(X^{ℓ}, P, P′, g_{P′}). The composition

e(X^{ℓ}, g_{P′}) ∘ ρ_{P,P′}^{X,u}

is defined by first parallel translating the vectors defined by the elements of Z_{P′} from y to the nearby points in the support of y′, and then exponentiating. Because, by (4.29) and the first assumption on the metrics g_{P} and g_{P′}, the g_{P,y} is flat where this parallel translation takes place, the two compositions are equal. □

4.4. Constructing the Thom-Mather maps. We now show how to construct the families of metrics which satisfy the conditions of Lemma 4.8 and thus yield projection maps π(X^{ℓ}, g_{P}) which satisfy the first condition of (4.1). The construction of tubular distance functions is then quite straightforward.

4.4.1. The projection maps. The maps π(X^{ℓ}, g_{P}) were defined in the previous section. For P < P′ partitions of N_{ℓ}, the first Thom-Mather property

(4.30)  π(X^{ℓ}, g_{P}) ∘ π(X^{ℓ}, g_{P′}) = π(X^{ℓ}, g_{P})

will follow from the commutativity of the diagram (4.23). Explicitly: Lemma 4.9. Continue the hypotheses and notation of Lemma 4.8. Then, restricted to Ũ′ ∩ Δ^{∘}(X^{ℓ}, P′), the equality (4.30) holds.

Proof. The subspace Ũ′ ∩ Δ^{∘}(X^{ℓ}, P′) is in the image of the composition e(X^{ℓ}, g_{P}) ∘ ρ_{P,P′}^{X,d}. The equality follows from the observation that the pullback of π(X^{ℓ}, g_{P′}) by ρ_{P,P′}^{X,d} is given by the projection π_{1} of (4.27). □

Thus, we must go about constructing families of metrics g_{P} for each partition of N_{ℓ} which satisfy the conditions of Lemma 4.8. The first step is showing how to flatten a family of metrics.

Lemma 4.10. If g_{P} is a family of metrics smoothly parameterized by Δ^{∘}(X^{ℓ}, P), then there is a family of smooth metrics F (g_{P}) smoothly parameterized by Δ^{∘}(X^{ℓ}, P) such that

(1) F (g_{P}) is locally flat with respect to a tubular neighborhood Ũ′ ⊆ Ũ(X^{ℓ}, P, g_{P}),

(2) For every x ∈ Δ^{∘}(X^{ℓ}, P), F (g_{P})_{x} is equal to g_{P,x} on the complement of the support of Ũ(X^{ℓ}, P, g_{P}) ∩ π(X^{ℓ}, g_{P})^{−1}(x),

(3) g_{P} and F (g_{P}) are C^{1} close within support of Ũ′,

(4) The exponential maps e(X^{ℓ}, g_{P}) and e(X^{ℓ}, F (g_{P})) are equal.

(5) If g_{P} is already flat with respect to Ũ(X^{ℓ}, P, g_{P}) at x ∈ Δ^{∘}(X^{ℓ}, P), then F (g_{P,x}) = g_{P,x}.

Proof. Let Ũ(X^{ℓ}, P, g_{P}) be the tubular neighborhood of Δ^{∘}(X^{ℓ}, P) defined by the family of metrics g_{P}, so e(X^{ℓ}, g_{P}) : Õ(X^{ℓ}, P, g_{P}) ⊂ ν̃(X^{ℓ}, P) ≃ Ũ(X^{ℓ}, P, g_{P}). For P ∈ P, let s_{P} : Δ^{∘}(X^{ℓ}, P) → [0, ∞) be a smooth function such that (4.31)

{(v_{1}, …, v_{ℓ}) ∈ ν̃(X^{ℓ}, P)|_{x} : |v_{i}| ≤ 4s_{P} (x) for all i ∈ P and for all P ∈ P} ⊂ O(P)

For x ∈ Δ^{∘}(P), let (F_{P})_{P ∈P} be frames in Fr(T X^{ℓ}, P, g_{P}) lying over x. Use the frame F_{P} and the exponential map of the metric g_{P,x} to identify B(0, 4s_{P} (x)) ⊂ ℝ^{4} with B_{g_{P,x}}(x_{P} , 4s_{P} (x)). Let δ_{x_{P}} denote the flat metric on B_{g_{P,x}}(x_{P} , 4s_{P} (x)) given by the pushforward of the Euclidean metric by the exponential map. Let χ : ℝ → ℝ satisfy: χ(x) = 0 for x ≤ \frac{1}{2} and χ(x) = 1 for x ≥ 1. Then, we define F (g)_{P,x} by: (4.32)

F (g)_{P,x} = \begin{cases}g_{P,x} & on X \ ∪_{i}B_{g_{P,x}}(x_{P} , 4s_{P} (x)), \\ χ_{4s_{P} (x)}(|x|)g_{P,x} + (1 − χ_{4s_{P} (x)}(|x|))δ_{x_{P}} & on Ω_{g_{P,x}}(x_{P} , 2s_{P} (x), 4s_{P} (x))), \\ δ_{x_{P}} & on B_{g_{P,x}}(x_{P} , 2s_{P} (x)) \end{cases}

where x refers to the coordinates given by the g_{P,x} exponential map.

The resulting family F (g)_{P} is locally flat with respect to the tubular neighborhood Ũ′ defined by replacing 4s_{P} (x) with 2s_{P} (x) in (4.31).

Because the metric g_{P,x} has not been changed at x_{P}, the frame bundles do not change. If the metric g_{P,x} is flat on B(x_{P} , 4s_{P} (x)), then it is given by the Euclidean metric in exponential coordinates, by [26, Vol. II, Ch. 7, Thm. 11, p. 306]. Thus, δ_{x_{P}} = g_{P,x} and the interpolation in (4.32) does not change the metric. □

We now construct the desired families of metrics. Lemma 4.11. For each partition P of N_{ℓ}, there is a smooth family of metrics on X, g_{P}, defining tubular neighborhoods Ũ(X^{ℓ}, P, g_{P}) by their exponential maps, such that

(1) The family g_{P} is locally flat with respect to a tubular neighborhood Ũ′(X^{ℓ}, P, g_{P}) ⊂ Ũ(X^{ℓ}, P, g_{P}),

(2) If P < P′, then for all x′ ∈ Δ^{∘}(X^{ℓ}, P′) ∩ Ũ′(X^{ℓ}, P, g_{P}) g_{P′,x′} = g_{P,x} where x = π(X^{ℓ}, g_{P})(x′).

Proof. The proof is by induction on the partitions P.

Fix a metric g on X. To every partition P of N_{ℓ}, define an initial family of metrics g_{P} by g_{P,x} = g.

Let P_{0} be the crudest partition. Redefine g_{P_{0}} by applying the flattening procedure of Lemma 4.10. Then the redefined family of metrics g_{P_{0}} is locally flat with respect to the tubular neighborhood Ũ′(X^{ℓ}, P_{0}, g_{P_{0}}) given in Lemma 4.10.

Assume families of metrics g_{P} and tubular neighborhoods Ũ′(X^{ℓ}, P, g_{P}) satisfying the conclusions of the Lemma have been defined for all P < P′. There are two families of metrics on X parameterized by the intersection

(4.33)  Δ^{∘}(X^{ℓ}, P′) ∩ U(X^{ℓ}, P, g_{P}).

There is the initial family g_{P′} and the family π(X^{ℓ}, g_{P})^{∗}g_{P} which is defined by

π(X^{ℓ}, g_{P})^{∗}g_{P,x′} = g_{P,x} where π(X^{ℓ}, g_{P})(x′) = x.

Observe that, by shrinking the tubular neighborhoods if necessary, for P_{1} < P′ and P_{2} < P′ if the intersection

(4.34)  Δ^{∘}(X^{ℓ}, P′) ∩ Ũ(X^{ℓ}, P_{1}, g_{P_{1}}) ∩ Ũ(X^{ℓ}, P_{2}, g_{P_{2}})

is non-empty, then there is a relation P_{1} < P_{2} < P′. Then, the relations

π(X^{ℓ}, P_{1}) ∘ π(X^{ℓ}, P_{2}) = π(X^{ℓ}, P_{1})

(which follows by induction and Lemma 4.9) and

g_{P_{2}} = π(X^{ℓ}, P_{1})^{∗}g_{P_{1}}

imply that the families of metrics

(4.35)  π(X^{ℓ}, g_{P_{1}})^{∗}g_{P_{1}} and π(X^{ℓ}, g_{P_{1}})^{∗}g_{P_{1}}

are equal on (4.34). Thus we can form a new family of metrics by interpolating between g_{P′} on

Δ^{∘}(X, P′) − ∪_{P<P′} Ũ′(X^{ℓ}, P, g_{P}),

and the pullback metrics (4.35) . By further shrinking the tubular neighborhoods Ũ′(X^{ℓ}, P, g_{P}) for P < P′ and taking a sufficiently small neighborhood Ũ′(X^{ℓ}, P′, g_{P′}) we can assume this interpolated family is locally flat with respect to Ũ′(X^{ℓ}, P′, g_{P′}) on

(4.36)  Δ^{∘}(X^{ℓ}, P′) ∩ (∪_{P<P′} Ũ(X^{ℓ}, P, g_{P})) .

Then, apply the flattening procedure of Lemma 4.10 to this interpolated family. Because this family is already flat with respect to Ũ′(X^{ℓ}, P′, g_{P′}) for x in the open space (4.36), the flattening procedure does not change the interpolated family there. Thus, the new, flattened family satisfies both properties of the conclusion of the lemma, completing the induction. □

4.4.2. The tubular distance function. Given the family of metrics constructed in the previous section, the tubular distance functions t(P) are defined by the composition of a bundle norm on ν̃(X^{ℓ}, P) with the inverse of the exponential map e(X^{ℓ}, g_{P})^{−1}. These do not satisfy the second Thom-Mather identity, (4.1), immediately.

We define a new tubular distance function t(X^{ℓ}, P) and a rescaling of it t_{r}(X^{ℓ}, P) on a possibly smaller tubular neighborhoods. These new functions will satisfy (4.1). Lemma 4.12. For P a partition of N_{ℓ}, let g_{P} be the smooth family of metrics parameterized by Δ^{∘}(X^{ℓ}, P) and let Ũ′(X^{ℓ}, P, g_{P}) be the tubular neighborhood of Δ^{∘}(X^{ℓ}, P) constructed in Lemma 4.11. Then there are tubular neighborhoods Ũ′′(X^{ℓ}, P, g_{P}) and 𝔖(P)-invariant smooth functions

(4.37)  t(X^{ℓ}, P) : Ũ′′(X^{ℓ}, P, g_{P}) → [0, 1], and t_{r}(X^{ℓ}, P) : Ũ′′(X^{ℓ}, P, g_{P}) → [0, 1],

such that (1) t_{r}(X^{ℓ}, P) is a rescaling of t(X^{ℓ}, P),

(2) t(X^{ℓ}, P)^{−1}(0) = t_{r}(X^{ℓ}, P)^{−1}(0) = Δ^{∘}(X^{ℓ}, P), (3) t_{r}(X^{ℓ}, P)^{−1}([0, \frac{1}{2})) ⋐ Ũ′′(X^{ℓ}, P, g_{P})

(4) For all partitions P < P′, the equalities

(4.38)  \begin{aligned}t(X^{ℓ}, P) ∘ π(X^{ℓ}, P′, g_{P′}) & = t(X^{ℓ}, P), \\ t_{r}(X^{ℓ}, P) ∘ π(X^{ℓ}, P′, g_{P′}) & = t_{r}(X^{ℓ}, P), \end{aligned}

hold on U′′(X^{ℓ}, g_{P}) ∩ U′′(X^{ℓ}, P′, g_{P′}).

Proof. Because the functions t(Z_{P}) : Z_{P} → [0, ∞) defined in Lemma 4.3 are invariant under the SO(4) action on Z_{P}, the function ∑_{P ∈P} |P |t(Z_{P})^{2} is defined on

Fr(T X^{ℓ}, P, g_{P} ×_{G(T,P)} ∏_{P ∈P} Z_{P} ,

and hence

(4.39)  t(X^{ℓ}, P) = ( ∑_{P ∈P} |P |t(Z_{P})^{2}) ∘ e(X^{ℓ}, g_{P})^{−1}

is defined on Ũ′(X^{ℓ}, P, g_{P}) and satisfies item (2) of the lemma. By definition, t(X^{ℓ}, P) is invariant under the action of 𝔖(P). By the construction of the tubular neighborhoods Ũ′(X^{ℓ}, P, g_{P}) in Lemma 4.11, for P < P′ the intersection Ũ′(X^{ℓ}, P, g_{P}) ∩ Ũ′(X^{ℓ}, P′, g_{P′}) is contained in the image of the neighborhood Õ(X^{ℓ}, P, P′, g_{P}) under the maps in the diagram (4.23). Under the identification of Ũ′(X^{ℓ}, P, g_{P}) ∩ Ũ′(X^{ℓ}, P′, g_{P′}) with a subspace of

Fr(T X^{ℓ}, P, g_{P}) ×_{G(T,P)} ∏_{P ∈P} (Δ^{∘}(Z_{P} , P′_{P}) × ∏_{P′∈P′_{P}} Z_{P′} )

(see (4.22)), the projection map π(X^{ℓ}, P′, g_{P′}) is identified with the projection

Fr(T X^{ℓ}, P, g_{P}) ×_{G(T,P)} ∏_{P ∈P} (Δ^{∘}(Z_{P} , P′_{P}) × ∏_{P′∈P′_{P}} Z_{P′})
↓
Fr(T X^{ℓ}, P, g_{P}) ×_{G(T,P)} ∏_{P ∈P} Δ^{∘}(Z_{P} , P′_{P})

By item (2) in Lemma 4.3, by the definition of the map ρ_{P,P′}^{X,d}, and by the definition of t(X^{ℓ}, P) in (4.39) as a sum of the functions t(Z_{P}), the function t(X^{ℓ}, P) satisfies (4.38).

We define t_{r}(X^{ℓ}, P) to be a rescaling of t(X^{ℓ}, P). The function f : Δ^{∘}(X^{ℓ}, P) used to rescale t(X^{ℓ}, P) is chosen to ensure item (3) holds. Then, the equality

t_{r}(X^{ℓ}, P)(·) = t(X^{ℓ}, P)(·)f(π(X^{ℓ}, P, g_{P})(·)),

and the equality

f ∘ π(X^{ℓ}, P, g_{P}) = f ∘ π(X^{ℓ}, P, g_{P}) ∘ π(X^{ℓ}, P′, g_{P′})

(which follows from (4.30)), and the fact that (4.38) holds for t(X^{ℓ}, P) ensure that t_{r}(X^{ℓ}, P) also satisfies (4.38). □

Remark 4.13. All the constructions of this section can be made 𝔖_{ℓ} equivariant and thus descend to the symmetric product Sym^{ℓ}(X).

5. The instanton moduli space with spliced ends

The first step in constructing the space of global splicing data is to define the moduli space with spliced ends. This moduli space will be identical to M̄_{κ}^{s,♮}(δ), the Uhlenbeck compactification of the mass-centered, charge κ, framed instantons on ℝ^{4} with scale less than δ, except on an Uhlenbeck neighborhood of the punctured, centered symmetric product,

(5.1)  [Θ] × (Sym_{δ}^{κ,♮}(ℝ^{4}) \ c_{κ})

where Θ is the trivial connection, Sym_{δ}^{κ,♮}(ℝ^{4}) is the subspace of mass-centered points in the κ-th symmetric product of ℝ^{4} with a scale constraint (see (5.4)), and c_{κ} ∈ Sym^{κ}(ℝ^{4}) is the point totally concentrated at the origin.

A neighborhood of the strata (5.1) in M̄_{κ}^{s,♮}(δ) is parameterized by the union of the images of the gluing maps γ_{Θ,Σ}, as Σ varies over the strata in (5.1). Roughly speaking, if the stratum Σ of Sym^{κ}(ℝ^{4}) is given by the partition κ = κ_{1} + ⋯ + κ_{r}, and P is a partition of N_{κ} corresponding to Σ then the gluing map is an smoothly stratified embedding

γ_{Θ,P} : Σ × ∏_{i=1}^{r} M̄_{κ_{i}}^{s,♮}(δ_{i}) → M̄_{κ}^{s,♮}(δ).

The scale parameters on the moduli spaces M̄_{κ_{i}}^{s,♮}(δ) make the image of γ_{Θ,Σ} into a kind of tubular neighborhood of {[Θ]} × Σ in M̄_{κ}^{s,♮}(δ). The moduli space with spliced ends will be defined, roughly speaking, by replacing the image of γ_{Θ,Σ} with the image of the splicing map γ′_{Θ,Σ} on the interior of this tubular neighborhood. We summarize the properties of the spliced-ends moduli space in the following theorem. Theorem 5.1. Let B̄_{κ}^{s}(δ) be the Uhlenbeck extension of the framed quotient space defined in (5.17). There exists a smoothly-stratified moduli space with spliced ends, M̄_{spl,κ}^{s,♮}(δ) ⊂ B̄_{κ}^{s}(2δ), closed under the SO(3) × SO(4) action, satisfying the following properties

(1) There is a smoothly-stratified, SO(3)×SO(4)-equivariant, homeomorphism, M̄_{spl,κ}^{s,♮}(δ) ≅ M̄_{κ}^{s,♮}(δ), which is the identity on the trivial strata,

[Θ] × Sym_{δ}^{κ,♮}(ℝ^{4}),

where Sym_{δ}^{κ,♮}(ℝ^{4}) ⊂ Sym^{κ}(ℝ^{4}) is defined in (5.4).

(2) There is an Uhlenbeck neighborhood W_{κ} of the punctured centered symmetric product, (5.1), in B̄_{κ}^{s}(2δ) such that

M̄_{spl,κ}^{s,♮}(δ) \ W_{κ} = M̄_{κ}^{s,♮}(δ) \ W_{κ}.

(3) To each stratum {[Θ]}×Σ in the punctured, centered symmetric product, (5.1), there is an Uhlenbeck neighborhood W (Σ) ⊂ B̄_{κ}^{s}(δ) of the trivial stratum

(5.2)  [Θ] × Σ,

and an open neighborhood O_{1}^{asd}(Θ, P, δ) ⊂ ν(Θ, P, δ) of the trivial stratum (5.53) in the domain of the splicing map γ′_{Θ,P} defined in (5.19), such that

(5.3)  M̄_{spl,κ}^{s,♮}(δ) ∩ W (Σ) = γ′_{Θ,P} (O_{1}^{asd}(Θ, P, δ)) .

(4) For all [A, x] ∈ M̄_{spl,κ}^{s,♮}, ∥F_{A}^{+}∥_{L^{♯,2}} ≤ M where M is the constant appearing in [7, Proposition 7.6].

We construct M̄_{spl,κ}^{s,♮}(δ) using induction on κ. We first define the spliced end of the moduli space. The spliced end will be an Uhlenbeck neighborhood of the punctured, centered symmetric product in M̄_{spl,κ}^{s,♮}(δ). It is defined as the union of the images of splicing maps γ′_{Θ,Σ} where Σ is a stratum in Sym_{δ}^{κ,♮}(ℝ^{4}) − {c_{κ}} and where the domain of the splicing map is not defined by the lower charge moduli spaces M̄_{κ_{i}}^{s,♮}(δ_{i}) but rather by the lower charge spliced ends moduli spaces M̄_{κ_{i}}^{s,♮}(δ_{i})(δ_{i}). A technical result on the equality of two splicing maps, Proposition 5.12 controls the overlaps of the images of the different splicing maps, proving that this union is a smoothly stratified space.

In §5.1 we introduce some vocabulary to describe the trivial strata. In §5.2, we define the relevant splicing maps. In §5.3, we prove the technical result Proposition 5.12 controlling the overlap of two splicing maps. Then in §5.4, we finish the first stage of the proof of Theorem 5.1 by constructing the spliced end in Proposition 5.13.

The second stage of the proof of Theorem 5.1 appears in §5.6. There, we show how to isotope the complement of a neighborhood of the punctured, centered symmetric product in the spliced end to the actual moduli space M̄_{κ}^{s,♮}(δ). This isotopy is defined by the composition of the isotopy defining the gluing map and the isotopy defined by the centering map. That such an isotopy exists follows immediately from the estimates on ∥F_{A}^{+}∥_{L^{♯,2}} for A in the spliced end; most of the work in §5.6 lies in constructing the parameter on the spliced end for the isotopy. The existence of this parameter is established in §5.5 by constructing a partial Thom-Mather structure on the spliced end.

5.1. The trivial strata. We begin with some vocabulary, mostly adapted from §3 to describe the trivial strata (5.1).

The symmetric product, Sym^{κ,♮}(ℝ^{4}) is the quotient of the zero locus of the center of mass map z_{κ} : ⊕_{i∈N_{κ}}ℝ^{4} → ℝ^{4} (defined following (4.3)) by the symmetric group:

Sym^{κ,♮}(ℝ^{4}) = z_{κ}^{−1}(0)/𝔖_{κ}.

We shall abbreviate z_{κ}^{−1}(0) by Z_{κ}. We define a scale for elements of Sym^{κ}(ℝ^{4}) by

(5.4)  λ([v_{1}, …, v_{κ}])^{2} = ∑_{i} |v_{i}|^{2},

Then, we define Z_{κ}(δ) = z_{κ}^{−1}(0) ∩ λ^{−1}([0, δ)) and

(5.5)  Sym_{δ}^{κ,♮}(ℝ^{4}) = Z_{κ}(δ)/𝔖_{κ}.

We describe the strata of Sym_{δ}^{κ,♮}(ℝ^{4}) by the quotients of the diagonals of Z_{κ}(δ). For any partition P of N_{κ}, define

(5.6)  \begin{aligned}Δ^{∘}(Z_{κ}(δ), P) & = Δ^{∘}(Z_{κ}, P) ∩ Z_{κ}(δ), \\ Σ(Z_{κ}(δ), P) & = Δ^{∘}(Z_{κ}(δ), P)/𝔖(P) ⊂ Sym_{δ}^{κ,♮}(ℝ^{4}) \end{aligned}

where Δ^{∘}(Z_{κ}, P) is defined in (4.7). Let Σ(Z_{κ}(δ), P) be the image of Δ^{∘}(Z_{κ}(δ), P) under the projection Z_{κ}(δ) → Sym_{δ}^{κ,♮}(ℝ^{4}). We then have:

Lemma 5.2. If P and P′ are partitions of N_{κ} and π_{κ} : Z_{κ} ⊂ ⊕_{i∈N_{κ}}ℝ^{4} → Sym^{κ,♮}(ℝ^{4}) is the projection, then:

(1) Σ(Z_{κ}(δ), P) = Δ^{∘}(Z_{κ}(δ), P)/𝔖(P),

(2) π_{κ}^{−1} (Σ(Z_{κ}(δ), P)) = ⊔_{P′∈[P]}Δ^{∘}(Z_{κ}(δ), P′).

(3) Σ(Z_{κ}(δ), P) ⊂ cl (Σ(Z_{κ}(δ), P′)) if and only if there is P′′ ∈ [P] such that P′′ < P.

If (v_{1}, …, v_{κ}) ∈ Δ^{∘}(Z_{κ}, P) and P ∈ P, then v_{i} = v_{j} for i, j ∈ P. Thus, we will write v_{P} := v_{i} for any i ∈ P. With this notation, we will write (y_{1}, …, y_{κ}) = (y_{P})_{P ∈P} for an element of Δ^{∘}(Z_{κ}, P).

5.1.1. Tubular neighborhoods. A tubular neighborhood of a diagonal in Z_{κ}(δ) can be obtained using the exponential map with respect to the product of the flat metrics on ⊕_{i∈N_{κ}}ℝ^{4}: Lemma 5.3. The normal bundle of the diagonal Δ^{∘}(Z_{κ}(δ), P) in Z_{κ} is

(5.7)  \begin{aligned}ν(Z_{κ}(δ), P) & = \{((x_{1}, …, x_{κ}), (v_{1}, …, v_{κ})) ∈ Δ^{∘}(Z_{κ}(δ), P) × ⊕_{i∈N_{κ}}ℝ^{4} : \\ & for all P ∈ P ∑_{i∈P} v_{i} = 0\}. \end{aligned}

There is an open neighborhood, Õ(Z_{κ}(δ), P), of the zero section in ν(Z_{κ}(δ), P) such that the restriction of the exponential map e(Z_{κ}, P) defined by

(5.8)  e(Z_{κ}, P) ((x_{1}, …, x_{κ}), (v_{1}, …, v_{κ})) = (x_{1} + v_{1}, …, x_{κ} + v_{κ})

to Õ(Z_{κ}(δ), P) is injective and 𝔖(P) equivariant.

Let Ũ(Z_{κ}(δ), P) be the image of Õ(Z_{κ}(δ), P) under the exponential map (5.8). By the 𝔖(P) equivariance of the exponential map e(Z_{κ}(δ), P), there is a homeomorphism,

(5.9)  O(Z_{κ}(δ), P) ≅ U(Z_{κ}(δ), P)

where O(Z_{κ}(δ), P) = Õ(Z_{κ}(δ), P)/𝔖(P) and U(Z_{κ}(δ), P) = Ũ(Z_{κ}(δ), P)/𝔖(P).

To describe the pre-image of Δ^{∘}(Z_{κ}(δ), P′) under e(Z_{κ}, P) where P < P′, we introduce the following notation. First, for P ⊆ N_{κ} let Z_{P} ⊂ ⊕_{i∈P} ℝ^{4} be the mass-centered subspace defined in (4.3). Lemma 5.4. Let P < P′ be partitions of N_{κ}. For P ∈ P, let P′_{P} be the partition of P defined in (4.6). Then there is an inclusion

(5.10)  ν(Z_{κ}(δ), P, P′) = Δ^{∘}(Z_{κ}(δ), P) × ∏_{P ∈P} Δ^{∘}(Z_{P} , P′_{P}) → ν(Z_{κ}(δ), P)

such that if we define

(5.11)  Õ(Z_{κ}(δ), P, P′) = Õ(Z_{κ}(δ)(δ), P) ∩ ν(Z_{κ}(δ), P, P′),

then Õ(Z_{κ}(δ), P, P′) = e(Z_{κ}, P)^{−1}(Δ^{∘}(Z_{κ}(δ), P′)).

Proof. We observe that

Δ^{∘}(Z_{P} , P′_{P}) = {(v_{i})_{i∈P} ∈ Z_{P} : v_{i} = v_{j} if and only if there is P′ ∈ P′_{P} with i, j ∈ P′}.

Because P is a partition of N_{κ}, there is an isomorphism

⊕_{P ∈P} ⊕_{i∈P} ℝ^{4} ≃ ⊕_{i∈N_{κ}}ℝ^{4},

which induces an inclusion

(5.12)  ∏_{P ∈P} Δ^{∘}(Z_{P} , P′_{P}) → Z_{κ} ((v_{i})_{i∈P,P ∈P}) ↦ (v_{i})_{i∈N_{κ}}.

By the 𝔖(P) equivariance of the exponential map e(Z_{κ}, P), an element of the pre-image of Δ^{∘}(Z_{κ}(δ), P′) under e(Z_{κ}, P) can be written as

((x_{1}, …, x_{κ}), (v_{1}, …, v_{κ})) ∈ Δ^{∘}(Z_{κ}, P) × ⊕_{i∈N_{κ}}ℝ^{4}

where (v_{1}, …, v_{κ}) ∈ ⊕_{i∈N_{κ}}ℝ^{4} satisfies (1) ∑_{i} v_{i} = 0 for all P ∈ P,

(2) For all i, j, v_{i} = v_{j} if and only if there is P′ ∈ P′ with i, j ∈ P′.

Then one observes that the elements (v_{1}, …, v_{κ}) ∈ ⊕_{i∈N_{κ}}ℝ^{4} satisfying these two conditions are precisely the image of the inclusion (5.12). □

By the second and third items of Lemma 5.2, to describe the end of Σ(Z_{κ}(δ), P′) near Σ(Z_{κ}(δ), P), we must describe not only the end of Δ^{∘}(Z_{κ}(δ), P′) near Δ^{∘}(Z_{κ}(δ), P) as is done in Lemma 5.4, but also the ends of Δ^{∘}(Z_{κ}(δ), P′′) near Δ^{∘}(Z_{κ}(δ), P) for all P′′ ∈ [P′] with P < P′′.

Thus, define

(5.13)  ν(Z_{κ}(δ), [P < P′]) = Δ^{∘}(Z_{κ}(δ), P) × ∐_{P′′∈[P<P′]} ( ∏_{P ∈P} Δ^{∘}(Z_{P} , P′′_{P})) .

There is an action of 𝔖(P) on ν(Z_{κ}, [P < P′]) by the standard action on Δ^{∘}(Z_{κ}, P) and the action of 𝔖(P) on [P < P′]. Although Δ^{∘}(Z_{κ}(δ), P) is not a subset of ν(Z_{κ}(δ), [P < P′]), we use the phrase a neighborhood of Δ^{∘}(Z_{κ}(δ), P) in ν(Z_{κ}(δ), [P < P′]) to refer to the obvious parallel to subspaces of ν(Z_{κ}(δ), P). Lemma 5.5. Let P and P′ be partitions of N_{κ} with P < P′. Then a neighborhood of Σ(Z_{κ}(δ), P) in Σ(Z_{κ}(δ), P′) is homeomorphic to a neighborhood of the zero section in

ν(Z_{κ}(δ), [P < P′])/𝔖(P).

Denote this neighborhood by O(Z_{κ}(δ), [P < P′]).

Proof. The bundle ν(Z_{κ}(δ), [P < P′]) is the union of the bundles ν(Z_{κ}(δ), P, P′′) defined in (5.10) as P′′ varies in [P < P′] along the subspace Δ^{∘}(Z_{κ}(δ), P) of each of these bundles. Thus a neighborhood of Δ^{∘}(Z_{κ}(δ), P) in

(5.14)  ∪_{P′′∈[P<P′]}Δ^{∘}(Z_{κ}(δ), P′′)

is homeomorphic to a neighborhood of the zero-section in ν(Z_{κ}(δ), [P < P′]). By the characterization of the pre-image of Σ(Z_{κ}(δ), P) under the projection Z_{κ}(δ) → Sym_{δ}^{♮,κ}(ℝ^{4}) in Item (2) of Lemma 5.2, a neighborhood of Σ^{♮}(Z_{κ}(δ), P) in Σ^{♮}(Z_{κ}(δ), P′) is homeomorphic to the quotient of (5.14) by 𝔖(P), proving the lemma. □

5.2. Splicing to the trivial on ℝ^{4}. In this section, we define the basic operation of splicing as used in [9]. The main result is in Lemma 5.11 where we show that the composition of two splicing maps is equal to a single splicing map under suitably small open sets.

Let B_{κ} denote the quotient space of SU(2) connections on a bundle E_{κ} → S^{4} with c_{2}(E_{κ}) = κ. Let

B̄_{|P |} = ∪_{i=0}^{p} (B_{|P |−i} × Sym^{i}(ℝ^{4})) ,

denote the space of ideal connections on S^{4}, given the topology induced by Uhlenbeck convergence. We write elements of this space as [A, x] where [A] ∈ B_{|P |−ℓ} and x ∈ Sym^{ℓ}(ℝ^{4}). We define the center of mass z([A, x]), for x = (x_{1}, …, x_{ℓ}) by:

(5.15)  z ([A, x]) = \frac{1}{8π^{2}|P |} ∫_{ℝ^{4}} y|F_{A}|^{2}d^{4}y + ∑_{i=1}^{ℓ} x_{i}

and the scale, λ([A, x]) by

(5.16)  λ ([A, x])^{2} = \frac{1}{8π^{2}} ∫_{ℝ^{4}} |y − z([A, F^{s}])|^{2}|F_{A}|^{2}d^{4}y + ∑_{i=1}^{ℓ} |x_{i}|^{2}.

Define

B̄_{|P |}(ε) = λ^{−1}([0, ε]).

If [A, x] ∈ B̄_{|P |}(ε) and ε << 1, then the support of x is disjoint from the south pole. Thus, we can extend the framed quotient space B_{|P |}^{s}(ε) → B_{|P |}(ε) over the quotient space of ideal connections:

(5.17)  B̄_{|P |}^{s}(ε) = λ^{−1}([0, ε]) ∩ (⊔_{i=0}^{p} (B_{|P |−i}^{s} × Sym^{i}(ℝ^{4}))) .

We now define the operation of splicing connections to the trivial connection. Let P be a partition of N_{κ}. We define the splicing map on an open subspace of

Δ^{∘}(Z_{κ}(δ), P) ×_{𝔖(P)} ∏_{P ∈P} B̄_{|P |}^{s,♮}(δ_{P}),

where the constants δ_{P} are invariant under the action of 𝔖(P) and where 𝔖(P) acts on the generalized connections by permuting them, sending and element of B̄_{|P |}^{s,♮}(δ_{P}) to an

element of B̄_{|σ(P)|}^{s,♮}(δ_{|σ(P)|}). Note that elements of Δ^{∘}(Z_{κ}(δ), P) can be written as (y_{P})_{P ∈P},

where y_{P} ∈ ℝ^{4} while elements of ∏_{P ∈P} B̄_{|P |}^{s,♮}(δ_{P}) can be written as ([A_{P} , F_{P}^{s} , x_{P}])_{P ∈P}, where

[A_{P} , F_{P}^{s} , x_{P}] ∈ B̄_{|P |}^{s,♮}(δ_{P}). The open subspace on which the splicing map will be defined is

Õ(Θ, P, δ) := {(y_{P} , [A_{P} , F_{P}^{s} , x_{P}])_{P ∈P} ∈ Δ^{∘}(Z_{κ}(δ), P) × ∏_{P ∈P} B̄_{|P |}^{s}(δ_{P}) :

(5.18) for all P ≠ P′, 8\sqrt{λ([A_{P} , x_{P}])} + 8\sqrt{λ([A_{P′}, x_{P′})} < dist(x_{P} , x_{P′})\}.

Then, we define

(5.19)  γ′_{Θ,P} : O(ℝ^{4}, P) = Õ(ℝ^{4}, P)/𝔖(P) → B̄_{κ}^{s}(2δ),

as follows.

For P ∈ P, let E_{P} → S^{4} be the bundle supporting the connection A_{P}. The framed connection [A_{P} , F_{P}^{s}] defines a section of Fr(E_{P}), ϕ(A_{P} , F_{P}^{s}), over the complement of the north pole by parallel translation along great circles from the south pole. Let Θ(A_{P} , F_{P}^{s}) be the flat connection defined by this section. Note that if A_{P} is flat on the domain of ϕ(A_{P} , F_{P}^{s}), then Θ(A_{P} , F_{P}^{s}) = A_{P}.

If Θ denotes the trivial connection on ℝ^{4} × SO(3), then the flat connection Θ(A_{P} , F_{P}^{s}) is canonically identified with the restriction of Θ to the domain of the ϕ(A_{P} , F_{P}^{s}) as follows. The section ϕ(A_{P} , F_{P}^{s}) gives a trivialization of the bundle E and thus identifies it with the restriction of the trivial bundle ℝ^{4} × SO(3) to the domain of the section. Under this identification the connection Θ is identified with Θ(A_{P} , F_{P}^{s}).

We also note that given any two connections, A_{1} and A_{2}, a convex linear combination of them, tA_{1} + (1 − t)A_{2} also defines a connection. For example, if A_{1} is a connection which is trivial in a particular trivialization and A_{2} = A_{1} + a_{2}, we would write the connection one form for tA_{1} + (1 − t)A_{2} in this trivialization as (1 − t)a_{2}.

Let c_{y,λ} : ℝ^{4} → ℝ^{4} be defined by c_{λ,y}(z) = (z − y)/λ. Thus, c_{y,λ} maps the ball B(y, λ) to B(0, 1).

Let χ : ℝ → [0, 2) be a smooth function satisfying χ(x) = 0 for x ≤ \frac{1}{2} and χ(x) = 1 for x ≥ 1. Define χ_{y,λ} : ℝ^{4} → [0, 2) by χ_{y,λ}(x) = χ(|x − y|/λ).

Then, for y = (y_{P})_{P ∈P} ∈ Δ^{∘}(Z_{κ}(δ), P), λ_{P} = λ([A_{P} , F_{P}^{s} , x_{P}]), we define:

(5.20)  γ′_{Θ,P} (x, ([A_{P} , F_{P}^{s} , x_{P}])_{P ∈P}) = [A′, F_{P}^{s} , x′]

where the framed connection [A′, F_{P}^{s}] is defined by

A′ = \begin{cases}Θ & on ℝ^{4} \ ∪_{P ∈P}B(y_{P} , 4\sqrt{λ_{P}}) \\ (1 − χ_{y_{P} ,4\sqrt{λ_{P}}})c_{y_{P} ,1}^{∗}A_{P} + χ_{y_{P} ,4\sqrt{λ_{P}}} c_{y_{P} ,1}^{∗}Θ(A_{P} , F_{P}^{s}) & on Ω(y_{P} , 2\sqrt{λ_{P}} , 4\sqrt{λ_{P}})) \\ c_{y_{P} ,1}^{∗}A_{P} & on B(y_{P} , 2\sqrt{λ_{P}}). \end{cases} (5.21)

and the frame in [A′, F_{P}^{s}] is given by the canonical frame for the trivial connection Θ.

The point x′ in the expression (5.20) is defined by translating the singular support (and multiplicity) of the point x_{P} to c_{y_{P}} (x_{P}).

Note that we have not computed the behavior of the function λ ∘ γ′_{Θ,P} precisely so we cannot assert that the image of γ′_{Θ,P} is contained in B_{κ}^{s}(2δ). However, that containment will follow from the continuity of λ and of γ′_{Θ,P} with respect to Uhlenbeck limits if we shrink the domain O(Θ, P, δ) by requiring the scales λ_{P} to be sufficiently small.

We note the following result on the behavior of this splicing map at the cone point of B̄_{|P |}^{s}(δ_{P}).

Lemma 5.6. For any partition P of N_{κ}, the map γ′_{Θ,P} is continuous with respect to Uh-

lenbeck limits. If c_{|P |} ∈ Sym_{δ}^{|P |,♮}(ℝ^{4}) denotes the cone point, then for any (y_{P})_{P ∈P} ∈ Δ^{∘}(Z_{κ}(δ), P),

(5.22)  γ′_{Θ,P} ((y_{P})_{P ∈P}, ([Θ, c_{|P |})_{P ∈P})) = [Θ, (y_{p})_{P ∈P}]

It is important that our constructions in §5.4 be equivariant with respect to the SO(3) × SO(4) actions on the space of connections so that those constructions can be used in the space of gluing data which is defined with this action. Thus, we prove the following result on the equivariance of the splicing map. Lemma 5.7. Let P be a partition of N_{κ}. Let SO(3) × SO(4) act on

O(Θ, P, δ) ⊂ Δ^{∘}(Z_{κ}, P) ×_{𝔖(P)} ∏_{P ∈P} B̄_{|P |}^{s}(δ_{P}),

by, for A ∈ SO(3) and R ∈ SO(4),

(5.23)  ((x_{P})_{P ∈P}, [A_{P} , F_{P}^{s}]_{P ∈P}, A, R) → ((Rx_{P})_{P ∈P}, [(R^{−1})^{∗}A_{P} , RF_{P}^{s} A^{−1}]_{P ∈P}) ,

where R ∈ SO(4) acts on the frame by a lift to the principal bundle. Then, the splicing map

γ_{Θ,P} : O(Θ, P, δ) → B̄_{κ}^{s}(2δ),

is SO(3) × SO(4) equivariant.

Proof. The equivariance of the SO(4) action follows from noting that the inclusion

∐_{P ∈P} B(0, 4\sqrt{λ_{P}}) → ℝ^{4},

given by identifying B(0, 4\sqrt{λ_{P}}) with B(y_{P} , 4\sqrt{λ_{P}}) is SO(4) equivariant if R ∈ SO(4) acts on (x_{P})_{P ∈P} by

(x_{P})_{P ∈P} → (Rx_{P})_{P ∈P}.

(This follows because the action of SO(4) on a connection is given by taking any bundle map lift of the action on ℝ^{4}.) The equivariance of the SO(3) action follows from observing that splicing in A_{P} by identifying the section ϕ(A_{P} , F_{P}^{s}) with the identity section of ℝ^{4} × SO(3) gives the same connection as splicing in A_{P} by identifying the section ϕ(A_{P} , F_{P}^{s} A^{−1}) with the section x → (x, A^{−1}) of ℝ^{4} × SO(3). □

We will also require the following observation about centering maps. Lemma 5.8. Let c_{y,λ} : ℝ^{4} → ℝ^{4} be defined by c_{λ,y}(z) = (z − y)/λ. For any λ_{1}, λ_{2} > 0 and x_{1}, x_{2} ∈ ℝ^{4}, we have:

c_{x_{2},λ_{2}} ∘ c_{x_{1},λ_{1}} = c_{x_{1}+λ_{1}x_{2},λ_{1}λ_{2}},
c_{x_{2},λ_{2}}^{∗}χ_{x_{1},λ_{1}} = χ_{x_{2}+λ_{2}x_{1},λ_{1}λ_{2}}.

Proof. Elementary algebra. □

If one splices in a mass-centered connection of charge κ (to the trivial background) and scale λ at a point y ∈ ℝ^{4}, it is not immediately clear what the scale of the resulting connection will be. However, the limit of the scale of the spliced up connection (as λ goes to zero) will be \sqrt{κ|y|^{2}}. The following exploitation of that fact will be used in Lemma 5.11 to prove that, on a suitable open set of splicing data, when composing two splicing maps, the cutting-off operation of the second splicing map does not interact with that of the first splicing map. Lemma 5.9. Define T (Θ, P, δ) ⊂ O(Θ, P, δ) by

(5.24)  T (Θ, P, δ) = {(x_{P})_{P ∈P}, [Θ, c_{|P |}]_{P ∈P}) ∈ O(Θ, P, δ)}

Then,

(5.25)  γ′_{Θ,P}(T (Θ, P, δ)) = {[Θ]| × Σ(Z_{κ}(δ), P)

and there is an open neighborhood, D(Θ, P, δ) of T (Θ, P, δ) in O(Θ, P, δ) such that for every

A := (x_{P} , [A_{P} , F_{P}^{s} , x_{P}])_{P ∈P} ∈ D(Θ, P, δ),

and for every P ∈ P, if λ_{P} = λ([A_{P} , F_{P}^{s} , x_{P}]) and λ = λ(γ′_{Θ,P}(A)), then

(5.26)  B(x_{P} , 4λ_{P}^{1/2}) ⋐ B(0, 2λ^{1/2})

and

(5.27)  B(x_{P} , \frac{1}{4}λ_{P}^{1/3}) ⋐ B(0, \frac{1}{8}λ^{1/2}).

Proof. The equality (5.25) follows immediately from Lemma 5.6.

The functions λ_{P} vanish on T (Θ, P, δ) while λ ∘ γ′_{Θ,P} is equal to the square root of ∑_{P ∈P} |P ||x_{P} |^{2}, and thus non-zero. The continuity of the functions involved then implies the existence of the desired open subspace. □

5.3. Composing splicing maps. To construct the moduli space with spliced ends we need to show the images of the splicing maps γ′_{Θ,P} form a smoothly stratified space. Let P, P′ be partitions of N_{κ} with P < P′. For each P ∈ P, let P′_{P} be the partition of P defined in (4.6). We will need to understand the overlaps of the images of the maps γ′_{Θ,P} and γ′_{Θ,P′}. To that end, we define a space Õ(Θ, P, P′, δ) with maps to Õ(Θ, P, δ) and Õ(Θ, P′, δ) so that the following diagram commutes:

(5.28)  \begin{matrix}Õ(Θ, P, P′, δ) & ⟶^{ρ_{P,P′}^{Θ,d}} & Õ(Θ, P, δ) \\ ^{ρ_{P,P′}^{Θ,u}} ↓ & & ^{γ}′_{Θ,P} ↓ \\ Õ(Θ, P′, δ) & ⟶^{γ′_{Θ,P′}} & B̄_{κ}^{s}(2δ) \end{matrix}

We note that the d and u appearing in the maps ρ_{P,P′}^{Θ,d} and ρ_{P,P′}^{Θ,d} are supposed to imply “down” and “up” respectively as P and P′ correspond to the lower and upper strata.

5.3.1. Defining the overlap data. We now define the objects appearing in the diagram (5.28). The open set Õ(Θ, P, P′, δ) will be defined to be a subspace of

(5.29) ν(Θ, P, P′, δ) = Δ^{∘}(Z_{κ}(δ), P) × ∏_{P ∈P} (Δ^{∘}(Z_{|P |}(δ_{P}), P′_{P}) × ∏_{P′∈P′_{P}} B̄_{|P′|}^{s}(δ_{P′})) ,

where the constants δ_{P} and δ_{P′} will not need to be defined explicitly. We introduce the following projections from ν(Θ, P, P′, δ):

(5.30)

π_{P,x} : ν(Θ, P, P′, δ) → Δ^{∘}(Z_{κ}(δ), P),
π_{P′_{P} ,x} : ν(Θ, P, P′, δ) → Δ^{♮}(Z_{|P |}(δ_{P}), P′_{P}),
π_{x} : ν(Θ, P, P′, δ) → Δ^{∘}(Z_{κ}(δ), P) × ∏_{P ∈P} Δ^{♮}(Z_{|P |}(δ_{P}), P′_{P}),
π_{P} : ν(Θ, P, P′, δ) → Δ^{♮}(Z_{|P |}(δ_{P}), P′_{P}) × ∏_{P′∈P′_{P}} B̄_{|P′|}^{s},
π_{P′} : ν(Θ, P, P′, δ) → B̄_{|P′|}^{s}(δ_{P}).

Use the inclusion given in (5.10),

ν(Z_{κ}(δ), P, P′) = Δ^{∘}(Z_{κ}(δ), P) × ∏_{P ∈P} Δ^{∘}(Z_{P} (δ_{P}), P′_{P}) → ν(Z_{κ}(δ), P),

to consider ν(Z_{κ}(δ), P, P′) as a subspace of ν(Z_{κ}(δ), P). We may then apply the exponential map e(Z_{κ}, P) of (5.8) to elements of ν(Z_{κ}, P, P′). We desire to define the map ρ_{P,P′}^{Θ,u} appearing in the diagram (5.28) by

(5.31)  ρ_{P,P′}^{Θ,u} = (e(Z_{κ}, P) ∘ π_{x}) × ∏_{P′∈P′} π_{P′}.

That is, the map ρ_{P,P′}^{Θ,u} leaves the connection data in ν(Θ, P, P′, δ) alone and maps the points in ν(Z_{κ}(δ), P, P′) to the diagonal Δ^{∘}(Z_{κ}(δ), P′).

For the composition e(Z_{κ}, P) ∘ π_{x} in (5.31) to be defined, we must restrict the domain of ρ_{P,P′}^{Θ,u} to an open subspace Õ(Θ, P, P′, δ) of ν(Θ, P, P′, δ) satisfying

(5.32)  Õ(Θ, P, P′, δ) ⊂ π_{x}^{−1} (O(Z_{κ}(δ), P, P′)) ,

where O(Z_{κ}(δ), P, P′) is defined in (5.11).

To define the composition γ′_{Θ,P′}∘ρ_{P,P′}^{Θ,u} appearing in the diagram (5.28), the open subspace Õ(Θ, P, P′, δ) must satisfy

(5.33)  Õ(Θ, P, P′, δ) ⊂ (ρ_{P,P′}^{Θ,u})^{−1} (Õ(Θ, P′, δ)) .

Before defining ρ_{P,P′}^{Θ,d}, we note that if P ∈ P ∩ P′ then the partition P′_{P} is just the set P and Δ^{♮}(Z_{|P |}, P′_{P}) is a single point (the zero vector). In this case, we define the splicing map

γ′_{Θ,P′_{P}} : Δ^{♮}(Z_{|P |}(δ), P′_{P}) × B̄_{|P |}^{s}(δ_{P}) → B̄_{|P |}^{s}(2δ),

to be the projection onto B̄_{|P |}^{s}(δ_{P}) ⊂ B̄_{|P |}^{s}(2δ). Then, the map ρ_{P,P′}^{Θ,d} of (5.28) will be defined by:

(5.34)  ρ_{P,P′}^{Θ,d} := π_{P,x} × ( ∏_{P ∈P} γ′_{Θ,P′_{P}} ∘ π_{P} )

That is, the map ρ_{P,P′}^{Θ,d} leaves the point in Δ^{∘}(Z_{κ}, P) alone and for each P ∈ P applies the splicing map γ′_{Θ,P′_{P}} to the space

Δ^{∘}(Z_{|P |}(δ_{P}), P′_{P}) × ∏_{P′∈P′_{P}} B̄_{|P′|}^{s}(δ_{P′}).

For this map to be defined on Õ(Θ, P, P′, δ) we need the image of π_{P} to be contained in O(Θ, P′_{P} , δ_{P}), the domain of the splicing map γ′_{Θ,P′_{P}}. However, we will make a stronger requirement (to be used in Lemma 5.11):

(5.35)  Õ(Θ, P, P′, δ) ⊂ π_{P}^{−1} (D(Θ, P′_{P} , δ_{P})) for all P ∈ P \ P′,

where D(Θ, P′_{P} , δ_{P}) is the open subspace of Õ(Θ, P′_{P} , δ_{P}) defined in Lemma 5.9. Since D(Θ, P′_{P} , δ_{P}) ⊂ O(Θ, P′_{P} , δ_{P}), the condition (5.35) implies ρ_{P,P′}^{Θ,d} is defined on Õ(Θ, P, P′, δ).

Finally, for the composition γ′_{Θ,P}∘ρ_{P,P′}^{Θ,d} to be well defined, the open subspace Õ(Θ, P′_{P} , δ_{P}) must satisfy

(5.36)  Õ(Θ, P, P′, δ) ⊂ (ρ_{P,P′}^{Θ,d})^{−1} (Õ(Θ, P, δ)) .

We now record a result showing that the image of the restriction of γ′_{Θ,P′} ∘ ρ_{P,P′}^{Θ,u} to an appropriate open subspace Õ(Θ, P, P′, δ) of the overlap data ν(Θ, P, P′, δ) contains

(5.37)  {[Θ, x] : x ∈ e(Z_{κ}, P)(O(Z_{κ}(δ), P, P′))}.

Thus, the image of γ′_{Θ,P′} ∘ ρ_{P,P′}^{Θ,u} will contain all of the trivial stratum {[Θ]} × Σ(Z_{κ}(δ), P′) in the image of γ′_{Θ,P}. Lemma 5.10. Let P, P′ be partitions of N_{κ} satisfying P < P′. Let T (Θ, P, P′, δ) be the subspace of the space ν(Θ, P, P′, δ) given by

(5.38)  T (Θ, P, P′, δ) := π_{x}^{−1} (O(Z_{κ}, P, P′)) ∩ (∩_{P′∈P′}π_{P′}^{−1}([Θ, c_{|P′|}])) .

Then, there is an open subspace, Õ(Θ, P, P′, δ), of ν(Θ, P, P′, δ) satisfying the conditions (5.32),(5.33), (5.35), and (5.36) and also satisfying

(1) T (Θ, P, P′, δ) ⊂ Õ(Θ, P, P′, δ),

(2) γ′_{Θ,P′} ∘ ρ_{P,P′}^{Θ,u} (T (Θ, P, P′, δ)) = {[Θ, x] : x ∈ e(Z_{κ}, P) (O(Z_{κ}(δ), P, P′))},

(3) (γ′_{Θ,P})^{−1} ([Θ] × e(Z_{κ}, P)(O(Z_{κ}(δ), P, P′))) = ρ_{P,P′}^{Θ,d} (T (Θ, P, P′, δ)).

Proof. First,

(5.39)  π_{x}(T (Θ, P, P′, δ)) = O(Z_{κ}(δ), P, P′),

by definition so there is an open neighborhood of T (Θ, P, P′, δ) in ν(Θ, P, P′, δ) satisfying (5.32). Second, for every P′ ∈ P′,

π_{P′}(T (Θ, P, P′, δ)) = [Θ, c_{|P′|}]

so π_{P} (T (Θ, P, P′, δ)) ⊂ T (Θ, P′_{P} , δ_{P}) ⊂ D(Θ, P, δ) for all P ∈ P and there is an open neighborhood of T (Θ, P, P′, δ) in ν(ℝ^{4}, P, P′) satisfying (5.35).

The second item in the conclusions of the lemma follows from the definitions of the spaces and maps and the value of the splicing map given in (5.22). The second item implies that ρ_{P,P′}^{Θ,u}(T (Θ, P, P′, δ)) ⊂ Õ(Θ, P′, δ), so there is an open neighborhood of T (Θ, P, P′, δ) in ν(Θ, P, P′, δ) satisfying (5.33).

Next, equation (5.22) implies that

ρ_{P,P′}^{Θ,d}|_{T (Θ,P,P′,δ)} = π_{x}|_{T (Θ,P,P′,δ)}.

This equality and (5.39) imply the third item of the conclusions of the lemma. This third item implies that ρ_{P,P′}^{Θ,d}(T (Θ, P, P′, δ)) is contained in the domain of γ′_{Θ,P′}, O(Θ, P, δ), so there is an open neighborhood of T (Θ, P, P′, δ) in ν(Θ, P, P′, δ) satisfying (5.36).

By taking the intersection of these open neighborhoods of T (Θ, P, P′, δ) in ν(Θ, P, P′, δ), we obtained the desired open neighborhood Õ(Θ, P, P′, δ). □

5.3.2. Equality of splicing maps. We now are in a position to prove that the diagram (5.28) commutes. The key point is the restriction (5.35) on the domain Õ(Θ, P, P′, δ). This ensures that in the iterated splicing construction defining the composition γ′_{Θ,P} ∘ ρ_{P,P′}^{Θ,d} (first the splicings γ′_{Θ,P′_{P}} defining the map ρ_{P,P′}^{Θ,d} are performed, then the splicing γ′_{Θ,P} is performed), the cut-off function of the second splicing has no effect. This follows from Lemma 5.9 which ensures that the connections given by ρ_{P,P′}^{Θ,d} are already trivial on the annuli on which the splicing map γ′_{Θ,P} interpolates between them and the trivial connection. Thus, this iterated splicing procedure is equivalent to a single splicing, that given by the composition γ′_{Θ,P′} ∘ ρ_{P,P′}^{Θ,u}.

Lemma 5.11. Let P and P′ be partitions of N_{κ} with P < P′. Let Õ(Θ, P, P′, δ) be the open set defined in Lemma 5.10 of the space ν(Θ, P, P′, δ) defined in (5.29). Then, the following diagram commutes:

(5.40)  \begin{matrix}Õ(Θ, P, P′, δ) & ⟶^{ρ_{P,P′}^{Θ,u}} & Õ(Θ, P′, δ) \\ ^{ρ_{P,P′}^{Θ,d}} ↓ & & ^{γ}′_{Θ,P′} ↓ \\ Õ(Θ, P, δ) & ⟶^{γ′_{Θ,P}} & B̄_{κ}^{s}(2δ) \end{matrix}

Proof. To simplify the notation, we write Q for an element of the partition P′. Let A ∈ Õ(Θ, P, P′, δ) be given by:

A = ((y_{P})_{P ∈P}, ((x_{Q})_{P′∈P′_{P}} , [A_{Q}, F_{Q}^{s}]_{Q∈P′_{P}})) ,

where (y_{P})_{P ∈P} ∈ Δ^{∘}(Z_{κ}, P) and ((x_{Q})_{Q∈P′_{P}} , [A_{Q}, F_{Q}^{s}]_{Q∈P′_{P}}) ∈ D(Θ, P′_{P} , δ_{P}). (For simplicity of exposition, we assume [A_{Q}, F_{Q}^{s}] ∈ B̄_{|Q|}^{s}(δ_{Q}) has no ideal points; the proof is no more difficult without that assumption, but the notation becomes more opaque.)

The image of A under ρ_{P,P′}^{Θ,u} is given by the same connections, [A_{Q}, F_{Q}^{s}] as given in A, but by the point (x′_{Q})_{Q∈P′} ∈ Δ^{∘}(Z_{κ}(δ), P′), where

(5.41)  x′_{Q} = y_{P} + x_{Q} if Q ∈ P′_{P}.

(Note that x′_{Q} = y_{P} if Q ∈ P ∩ P′ as x_{Q} is then the zero vector.) If we set λ_{Q} = λ([A_{Q}]), then

γ′_{Θ,P′} ∘ ρ_{P,P′}^{Θ,u}(A) (5.42)

= \begin{cases}Θ & on ℝ^{4} \ ∪_{Q∈P′_{P} ,P ∈P}B(x′_{Q}, 4\sqrt{λ_{Q}}), \\ (1 − χ)c_{x′_{Q},4\sqrt{λ_{Q}}}^{∗}A_{Q} + χ_{x′_{Q},4\sqrt{λ_{Q}}}c_{x′_{Q}}^{∗}Θ(A_{Q}, f_{E_{Q}}) & on Ω(x′_{Q}, 2\sqrt{λ_{Q}}, 4\sqrt{λ_{Q}}), \\ c_{x′_{Q},1}^{∗}A_{Q} & on B(x′_{Q}, 2\sqrt{λ_{Q}}). \end{cases}

We now compare (5.42) with γ′_{Θ,P} ∘ ρ_{P,P′}^{Θ,d}(A).

If, for P ∈ P \ P′, we define

[A_{P} , F_{P}^{s}] = γ′_{Θ,P′_{P}} (π_{P} (A)) = γ′_{Θ,P′_{P}} ((x_{Q})_{Q∈P′_{P}} , [A_{Q}, f_{E_{Q}}]_{Q∈P′_{P}} ) ,

and [A_{P} , F_{P}^{s}] = [A_{Q}, F_{Q}^{s}] for P = Q ∈ P ∩ P′, then

ρ_{P,P′}^{Θ,d}(A) = ((y_{P})_{P ∈P}, ([A_{P} , F_{P}^{s}])_{P ∈P}) ∈ O(Θ, P, δ).

The composition γ′_{Θ,P} ∘ ρ_{P,P′}^{Θ,d} is then given by:

γ′_{Θ,P} ∘ ρ_{P,P′}^{Θ,d} (5.43)

= \begin{cases}Θ & on ℝ^{4} \ ∪_{P ∈P}B(y_{P} , 4\sqrt{λ_{P}}), \\ (1 − χ_{y_{P} ,4\sqrt{λ_{P}}})c_{y_{P} ,1}^{∗}A_{P} + χ_{y_{P} ,4\sqrt{λ_{P}}} c_{y_{P} ,1}^{∗}Θ(A_{P} , f_{E_{P}}) & on Ω(y_{P} , 2\sqrt{λ_{P}} , 4\sqrt{λ_{P}}), \\ c_{y_{P} ,1}^{∗}A_{P} & on B(y_{P} , 2\sqrt{λ_{P}}), \end{cases}

where λ_{P} = λ([A_{P}]). On the balls B(x_{P} , 4\sqrt{λ_{P}}) for P ∈ P∩P′, and on ℝ^{4}\∪_{P ∈P}B(x_{P} , 4\sqrt{λ_{P}}) the connections (5.42) and (5.43) are identical. We thus focus our attention on the balls B(x_{P} , 4\sqrt{λ_{P}}) for P ∈ P \ P′. For such P,

A_{P} = γ′_{Θ,P′_{P}} (π_{P} (A)) (5.44)

= \begin{cases}Θ & on ℝ^{4} \ ∪_{Q∈P′_{P}} B(x_{Q}, 4\sqrt{λ_{Q}}), \\ (1 − χ_{x_{Q},\sqrt{4λ_{Q}}})c_{x_{Q},1}^{∗}A_{Q} + χ_{x_{Q},\sqrt{4λ_{Q}}}c_{x_{Q},1}^{∗}Θ(A_{Q}, f_{E_{Q}}) & on Ω(x_{Q}, 2\sqrt{λ_{Q}}, 4\sqrt{λ_{Q}}), \\ c_{x_{Q},1}^{∗}A_{Q} & on B(z_{Q}, 2\sqrt{λ_{Q}}). \end{cases}

Because the points (x_{Q})_{Q∈P′_{P}} ∈ Δ^{♮}(ℝ^{4}, P′_{P}) and the connections [A_{Q}, f_{E_{Q}}] ∈ B̄_{|Q|}^{s} form a point in D(Θ, P′_{P} , δ_{P}) by the construction of Õ(Θ, P, P′, δ), Lemma 5.9 and equation (5.26) imply that for all Q ∈ P′_{P},

B(x_{Q}, 4\sqrt{λ_{Q}}) ⊂ B(0, 2\sqrt{λ_{P}}).

Thus, the construction in (5.44) implies A_{P} is already equal to Θ on

ℝ^{4} \ B(0, 2\sqrt{λ_{P}}),

so c_{y_{P} ,1}^{∗}A_{P} is equal to Θ on

ℝ^{4} \ B(y_{P} , 2\sqrt{λ_{P}}).

Thus, the convex combination in the second line of (5.43) is equal to Θ. We examine the final line of (5.43) by applying Lemma 5.8:

c_{y_{P} ,1}^{∗}A_{P} (5.45)

= \begin{cases}Θ & on ℝ^{4} \ ∪_{Q∈P′_{P}} B(y_{P} + x_{Q}, 4\sqrt{λ_{Q}}), \\ (1 − χ)c_{y_{P} +x_{Q},4\sqrt{λ_{Q}}}^{∗}A_{Q} + χ_{y_{P} +x_{Q},4\sqrt{λ_{Q}}}c_{y_{P} +x_{Q}}^{∗}Θ(A_{Q}, f_{E_{Q}}) & on Ω(y_{P} + x_{Q}, 2\sqrt{λ_{Q}}, 4\sqrt{λ_{Q}}), \\ c_{y_{P} +x_{Q},1}^{∗}A_{Q} & on B(y_{P} + x_{Q}, 2\sqrt{λ_{Q}}). \end{cases}

Thus, we see that γ′_{Θ,P} ∘ ρ_{P,P′}^{Θ,d}(A) is given by: (5.46)

Θ  on ℝ^{4} \ ∪_{Q∈P′_{P} ,P ∈P}B(y_{P} + x_{Q}, 4\sqrt{λ_{Q}}),
(1 − χ)c_{y_{P} +x_{Q},4\sqrt{λ_{Q}}}^{∗}A_{Q} + χ_{y_{P} +x_{Q},4\sqrt{λ_{Q}}}c_{y_{P} +x_{Q}}^{∗}Θ(A_{Q}, f_{E_{Q}}) on Ω(y_{P} + x_{Q}, 2\sqrt{λ_{Q}}, 4\sqrt{λ_{Q}}),
c_{y_{P} +x_{Q},1}^{∗}A_{Q}  on B(y_{P} + x_{Q}, 2\sqrt{λ_{Q}}).

Then comparing (5.46) with (5.42) and noting the definition of x_{Q} in (5.41) completes the proof. □

5.3.3. Symmetric group quotients. Finally, we make some observations on the action of the symmetric group necessary to define a quotient of the diagram (5.28) by the symmetric group. Recall from Lemma 5.5 that for P < P′, to describe a neighborhood of Σ(Z_{κ}(δ), P) in Σ(Z_{κ}, P), we must describe all the diagonals Δ^{♮}(Z_{κ}(δ), P′′) where P′′ ∈ [P < P′]. We define

(5.47)  ν(Θ, P, [P′], δ) = ⊔_{P′′∈[P<P′]}ν(Θ, P, P′′, δ)

The group 𝔖(P) acts on ν(Θ, P, [P′], δ) by the obvious permutation. Note that if σ ∈ 𝔖(P) and P′ ∈ P′, then it is possible that σ(P′) ∈ P′′ ∈ [P < P′] with P′′ ≠ P′, thus explaining the need for the additional partitions P′′ ∈ [P < P′]. Let O(Θ, P, [P′], δ) ⊂ ν(Θ, P, [P′], δ) be the analogue of the open subspace O(Θ, P, P′, δ) defined in Lemma 5.10. We define (analogously to the definition of T (Θ, P, P′, δ) in (5.38))

(5.48)  \begin{matrix}T (Θ, P, [P′], δ) = & \{((y_{P})_{P ∈P}, (x_{Q})_{Q∈P′′}, ([Θ, c_{|Q|}])_{Q∈P′′}) ∈ ν(Θ, P, [P′], δ) : \\ ((y_{P})_{P ∈P}, (x_{Q})_{Q∈P′′}) ∈ O(Z_{κ}(δ), [P < P′])\}, \end{matrix}

where O(Z_{κ}(δ), [P < P′]) is defined in Lemma 5.5.

The maps ρ_{P,P′′}^{Θ,u} for P′′ ∈ [P < P′] define a map

(5.49) ρ_{P,[P′]}^{Θ,u} : O(Θ, P, [P′], δ) → ν(Θ, [P < P′], δ) = ( ∐_{P′′∈[P<P′]} ν(Θ, P′′, δ)) /𝔖(P).

Similarly, the coproduct of the maps ρ_{P,P′′}^{Θ,d} is invariant under the action of 𝔖(P) and thus defines a map

(5.50)  ρ_{P,[P′]}^{Θ,d} : O(Θ, P, [P′], δ) → O(Θ, P, δ).

The coproduct of the splicing maps γ′_{Θ,P′′}, over P′′ ∈ [P < P′], is invariant under the action of 𝔖(P) and thus defines a map (5.51)

γ′_{Θ,P,[P′]} = ( ∐_{P′′∈[P<P′]} γ′_{Θ,P′′} ) /𝔖(P) : ( ∐_{P′′∈[P<P′]} O(ℝ^{4}, P′′, δ)) /𝔖(P) → B̄_{κ}^{s}(2δ),

which has the same image as γ′_{Θ,P′}. Lemma 5.11 then implies the following.

Proposition 5.12. Let P and P′ be partitions of N_{κ} with P < P′. Let O(Θ, P, P′, δ) be the open subspace defined following(5.47). Then, the following diagram commutes:

(5.52)  \begin{matrix}O(Θ, P, [P′], δ) & ⟶^{ρ_{P,[P′]}^{Θ,u}} & (∐_{P′′∈[P<P′]} O(ℝ^{4}, P′′)) /𝔖(P) \\ ^{ρ_{P,[P′]}^{Θ,d}}↓ & & ^{γ}′_{Θ,P,[P′]}↓ \\ O(Θ, P, δ) & ⟶^{γ′_{Θ,P}} & B̄_{κ}^{s}(2δ) \end{matrix}

where γ′_{Θ,[P<P′]} is defined in (5.51).

5.4. The spliced end. We now construct the spliced end, given by the union of the images of the splicing maps γ′_{Θ,P} restricted to appropriate subspaces of O(Θ, P, δ).

This construction is inductive. We therefore assume that the spliced ends moduli space M̄_{spl,|P |}^{s,♮}(δ) with the properties stated in Theorem 5.1 has already been constructed for |P | < κ. Note that because we only need to describe Uhlenbeck neighborhoods of the punctured trivial strata, we only need to use splicing maps γ′_{Θ,P} where |P| > 1. For a partition P of N_{κ} with |P| > 1, |P | < κ for all P ∈ P. Therefore, the inductive hypothesis suffices for this construction.

The splicing maps will be restricted to the following spaces (where the precise choice of the constants δ_{P} > 0 will not be relevant to the rest of the discussion)

(5.53)

Õ^{asd}(Θ, P, δ) = Õ(Θ, P, δ) ∩ (Δ^{∘}(Z_{κ}(2δ), P) × ∏_{P ∈P} M_{spl,|P |}^{s,♮}(δ_{P})) ,
O^{asd}(Θ, P, δ) = Õ^{asd}(Θ, P, δ)/𝔖(P).

Observe that T (Θ, P, δ) ⊂ Õ^{asd}(Θ, P, δ) and the dimension of O^{asd}(Θ, P, δ) is equal to that of M̄_{κ}^{s,♮}(δ).

The spliced end is constructed by the following proposition.

Proposition 5.13. Assume that the spliced end moduli space M̄_{spl,κ′}^{s,♮}(δ) with the properties enumerated in Theorem 5.1 has already been constructed. Then for every partition P of there is an SO(3) × SO(4)-invariant, open neighborhood O_{1}^{asd}(Θ, P, δ) of T (Θ, P, δ) in O^{asd}(Θ, P, δ) such that the space,

(5.54)  W_{κ} = ∪_{P} γ′_{Θ,P} (O_{1}^{asd}(Θ, P, δ)) ,

is a smoothly-stratified subspace of B̄_{κ}^{s}(δ) with every point [A, F^{s}, x] ∈ W_{κ} satisfying the estimate ∥F_{A}^{+}∥_{L^{♯,2}} ≤ M.

We prove Proposition 5.13 by using Proposition 5.12 to control the overlaps of the images of the splicing maps. The transition maps ρ_{P,[P′]}^{Θ,u} and ρ_{P,[P′]}^{Θ,d} will be restricted to the following subspace of O(Θ, P, [P′], δ)

(5.55)

Õ^{asd}(Θ, P, [P′], δ)

= Õ(Θ, P, [P′], δ)

∩ (Δ^{∘}(Z_{κ}(δ), P) × ∐_{P′′∈[P<P′]} ∏_{P ∈P} (Δ^{∘}(Z_{P} (δ_{P}), P′′_{P}) × ∏_{Q∈P′′_{P}} M̄_{spl,|Q|}^{s,♮}(δ_{Q}))) ,

O^{asd}(Θ, P, [P′], δ) = Õ^{asd}(Θ, P, [P′], δ)/𝔖(P).

(Compare the definition (5.47)). Observe that T (Θ, P, [P′], δ) ⊂ O^{asd}(Θ, P, [P′], δ) where T (Θ, P, [P′], δ) is defined in (5.48) because [Θ, c_{|Q|}] ∈ M̄_{spl,|Q|}^{s,♮}(δ_{Q}) for all |Q| < κ.

We must first verify that the transition maps ρ_{P,[P′]}^{Θ,u} and ρ_{P,[P′]}^{Θ,d} map O^{asd}(Θ, P, [P′], δ) to O^{asd}(Θ, P′, δ) and O^{asd}(Θ, P, δ) respectively. Lemma 5.14. Assume M̄_{spl,κ′}^{s,♮}(δ) has been constructed satisfying the properties enumerated in Theorem 5.1 for all κ′ < κ. Let T (Θ, P, [P′], δ) be the subspace of O^{asd}(Θ, P, [P′], δ) defined in (5.48). Then there is an open neighborhood O_{1}^{asd}(Θ, P, [P′], δ) of T (Θ, P, [P′], δ) in O^{asd}(Θ, P, [P′], δ) such that the restriction of ρ_{P,[P′]}^{Θ,d} to O_{1}^{asd}(Θ, P, [P′], δ) is an open embedding of O_{1}^{asd}(Θ, P, [P′], δ) into O^{asd}(Θ, P, δ).

Proof. Induction and the property (5.3) in Theorem 5.1 imply that for each P ∈ P \ P′, there is an open neighborhood O_{1}^{asd}(Θ, P′_{P} , δ_{P}) of T (Θ, P′_{P} , δ) in O^{asd}(Θ, P′_{P} , δ) such that

γ′_{Θ,P′_{P}} (O_{1}^{asd}(Θ, P′_{P} , δ_{P})) ⊂ M̄_{spl,|P |}^{s,♮}(δ_{P}).

Thus, if in the definition of O^{asd}(Θ, P, [P′], δ), (5.55), we replace the factor

Δ^{∘}(Z_{P} (δ_{P}), P′′_{P}) × ∏_{Q∈P′′_{P}} M̄_{spl,|Q|}^{s,♮}(δ_{Q})

with the open subspace

O_{1}^{asd}(Θ, P′_{P} , δ_{P}) ⊂ Δ^{∘}(Z_{P} (δ_{P}), P′′_{P}) × ∏_{Q∈P′′_{P}} M̄_{spl,|Q|}^{s,♮}(δ_{Q})

the resulting open subspace O_{1}^{asd}(Θ, P, [P′], δ) of O^{asd}(Θ, P, [P′], δ) will satisfy

(5.56)  ρ_{P,[P′]}^{Θ,d} (O_{1}^{asd}(Θ, P, [P′], δ)) ⊂ O^{asd}(Θ, P, δ).

(Note that we are not concerned with the subsets P ∈ P ∩ P′ because ρ_{P,P′}^{Θ,d} is the identity on these factors and thus they do not affect the inclusion (5.56).)

That the restriction of ρ_{P,[P′]}^{Θ,d} to O_{1}^{asd}(Θ, P, [P′], δ) is an open embedding into O^{asd}(Θ, P, δ) follows from the assumption that γ′_{Θ,P′_{P}} gives an open embedding of O_{1}^{asd}(Θ, P′_{P} , δ_{P}) into M̄_{spl,|P |}^{s,♮}(δ_{P}). □

We now prove an analogy of Lemma 5.14 for the map ρ_{P,[P′]}^{Θ,u}. First, we must define the appropriate range of ρ_{P,[P′]}^{Θ,u}, to take into account the appearance of the conjugate partitions P′′ ∈ [P < P′]. Let

(5.57) O^{asd}(Θ, [P < P′], δ) = ( ∐_{P′′∈[P<P′]} O^{asd}(Θ, P′′, δ)) /𝔖(P) ⊂ ν(Θ, [P < P′], δ),

where ν(Θ, [P < P′], δ) is defined in (5.49). Then, Lemma 5.15. Continue the assumptions and hypotheses of Lemma 5.14. Then the restriction of the map ρ_{P,[P′]}^{Θ,u} to O^{asd}(Θ, P, [P′], δ) is an open embedding into O^{asd}(Θ, [P < P′], δ).

Proof. The lemma follows immediately from the definition of ρ_{P,[P′]}^{Θ,u} because, being defined

largely by the exponential map e(Z_{κ}, P) (see (5.31)), the map ρ_{P,[P′]}^{Θ,u} does not change the connections in the domain O(Θ, P, [P′], δ). □

The following lemma shows the existence of suitably small, SO(3) × SO(4) invariant neighborhoods of T (Θ, P, δ). Lemma 5.16. For any subspace V of B̄_{κ}^{s}, let cl(V) denote the closure of V in the Uhlenbeck topology. If V is any set in B̄_{κ}^{s} with satisfying

cl(V) ∩ γ′_{Θ,P} (T (Θ, P, δ)) = ∅,

then there is an SO(3) × SO(4) × 𝔖(P) invariant neighborhood, O_{1}^{asd}(Θ, P, δ), of T (Θ, P, δ) in O^{asd}(Θ, P, δ) such that the intersection

cl(V) ∩ γ′_{Θ,P} (O_{1}^{asd}(Θ, P, δ))

is empty and the inclusion,

cl(V) ∩ cl (γ′_{Θ,P} (O_{1}^{asd}(Θ, P, δ))) ⊂ cl(V) ∩ cl (γ′_{Θ,P} (T (Θ, P, δ))) ,

is satisfied.

Proof. For A ∈ O^{asd}(Θ, P, δ) given by A = ((x_{P})_{P ∈P}, ([A_{P} , F_{P}^{s}])_{P ∈P}), let λ_{P} (A) = λ(A_{P}). For x = (x_{P})_{P ∈P} ∈ Δ^{∘}(Z_{κ}, P), define

V (x, P) := {(x, (A_{P})_{P ∈P}) ∈ (γ′_{Θ,P})^{−1} (cl(V))}

and λ_{V} : Δ^{∘}(Z_{κ}, P) → (0, ∞) by

λ_{V} (x) = min_{A∈V (x,P)} ∑_{P ∈P} λ_{P}^{2} (A).

By assumption λ_{V} is non-zero on Δ^{∘}(Z_{κ}(δ), P). Let f : Δ^{∘}(Z_{κ}(δ), P) → (0, ε) be a continuous function with f(x) < \frac{1}{2}λ_{V} (x), and

(5.58)  f ((x_{P})_{P ∈P}) < \frac{1}{2} min_{P_{1}≠P_{2}∈P} |x_{P_{1}} − x_{P_{2}}|.

Then we define the neighborhood O_{1}^{asd}(Θ, P, δ) by

O_{1}^{asd}(Θ, P, δ) := {A = (x, ([A_{P} , F_{P}^{s}])_{P ∈P}) ∈ O^{asd}(Θ, P, δ) : ∑_{P ∈P} λ_{P}^{2} (A) < f(x)}.

This set has the desired invariance and its image does not intersect cl(V).

If {A(α)}_{α=1}^{∞} ⊂ O_{1}^{asd}(Θ, P, δ), with

A(α) = [x(α), ([A_{P} (α), F_{P}^{s} (α)])_{P ∈P}],

where x(α) ∈ Δ^{∘}(Z_{κ}, P) is the splicing points of A(α) and if

lim_{α→∞} γ′_{Θ,P} (A(α)) ∈ cl(V),

then

lim_{α→∞} x(α) ∉ Δ^{∘}(Z_{κ}, P).

The inequality (5.58) then implies that

lim_{α→∞} f(x(α)) = 0,

so the condition

∑_{P ∈P} λ_{P}^{2} (A(α)) < f(x(α))

in the definition of O_{1}^{asd}(Θ, P, δ) implies that for all P ∈ P

lim_{α→∞} λ(A_{P} (α)) = 0,

so lim_{α→∞} A_{P} (α) = [Θ, c_{|P |}], giving the final conclusion of the lemma. □

We now construct the subspaces O_{1}^{asd}(Θ, P, δ) referred to in Proposition 5.13. We will construct these subspaces to be 𝔖_{κ} invariant in the sense that if P′′ ∈ [P], then the open subspaces O_{1}^{asd}(Θ, P, δ) and O_{1}^{asd}(Θ, P′′, δ) are related by the are identified by the natural action of 𝔖_{κ} on ∐_{P_{i}∈[P_{1}]} ν(Θ, P_{i}). Thus, defining O_{1}^{asd}(ℝ^{4}, P′) for one P′ ∈ [P′] suffices to define the space

(5.59) O_{1}^{asd}(Θ, [P < P′]) = ( ∐_{P′′∈[P<P′]} O_{1}^{asd}(Θ, P′′, δ)) 𝔖(P) ⊂ O^{asd}(Θ, [P < P′], δ),

where O^{asd}(Θ, [P < P′], δ) is defined in (5.57).

Lemma 5.17. Assume M̄_{spl,κ′}^{s,♮}(δ) has been constructed satisfying the properties enumerated in Theorem 5.1 for all κ′ < κ. Then, to every partition P of N_{κ} there is an open neighborhood O_{1}^{asd}(Θ, P, δ) of T (Θ, P, δ) in O^{asd}(Θ, P, δ), such that

(1) These neighborhoods are closed under the SO(3) × SO(4) action given in Lemma 5.23,

(2) If P_{1} and P_{2} are conjugate under the action of 𝔖_{κ}, then the neighborhoods O_{1}^{asd}(Θ, P_{1}, δ) and O_{1}^{asd}(Θ, P_{2}, δ) are identified by the natural action of 𝔖_{κ} on ∐_{P_{i}∈[P_{1}]} ν(Θ, P_{i}),

(3) If P ≮ P′ and P′ ≮ P, then

γ′_{Θ,P}(O_{1}^{asd}(Θ, P, δ)) ∩ γ′_{Θ,P′}(O_{1}^{asd}(Θ, P′, δ)) = ∅,

(4) If P < P′ then there is an open neighborhood O_{1}^{asd}(Θ, P, [P′], δ) of the space T (Θ, P, [P′], δ) in O^{asd}(Θ, P, [P′], δ) such that

(5.60)

γ′_{Θ,P} (O_{1}^{asd}(Θ, P, δ)) ∩ γ′_{Θ,P′} (O_{1}^{asd}(ℝ^{4}, P′))
⊆ γ′_{Θ,P′} (ρ_{P,[P′]}^{Θ,u}(O_{1}^{asd}(Θ, P, [P′], δ)))
= γ′_{Θ,P} (ρ_{P,[P′]}^{Θ,d}(O_{1}^{asd}(Θ, P, [P′], δ)))

Proof. By induction, assume the open sets O_{1}^{asd}(Θ, P, δ) have been defined for all [P] < P′. Note that the conclusion still holds if we shrink the sets O_{1}^{asd}(Θ, P, δ). We now construct O_{1}^{asd}(Θ, P′, δ).

It suffices to construct O_{1}^{asd}(Θ, P′, δ) for a single P′ ∈ [P′] because for another element, P′′ ∈ [P′], one can define O_{1}^{asd}(Θ, P′′, δ) to be the image of O_{1}^{asd}(Θ, P′, δ) under the action of 𝔖_{κ}.

If P ≮ P′ and P′ ≮ P, then we assume O_{1}^{asd}(Θ, P, δ) has already been constructed. If O_{1}^{asd}(Θ, P, δ) has not already been constructed, then this step will appear later when the induction reaches P. The relation P ≮ P′ and P′ ≮ P implies that

(5.61)  cl (γ′_{Θ,P} (T (Θ, P, δ))) ∩ γ′_{Θ,P′} (T (Θ, P′, δ)) = ∅,
(5.62)  γ′_{Θ,P} (T (Θ, P, δ))) ∩ cl (γ′_{Θ,P′} (T (Θ, P′, δ))) = ∅.

Lemma 5.16 and (5.61) imply there is an SO(3)×SO(4) invariant neighborhood O_{1}^{asd}(Θ, P′, δ) such that

cl (γ′_{Θ,P} (T (Θ, P, δ))) ∩ γ′_{Θ,P′} (O_{1}^{asd}(Θ, P′, δ)) = ∅.

Lemma 5.16 also implies that

cl (γ′_{Θ,P} (T (Θ, P, δ))) ∩ cl (γ′_{Θ,P′} (O_{1}^{asd}(Θ, P′, δ)))

(5.63) ⊆ cl (γ′_{Θ,P} (T (Θ, P, δ))) ∩ cl (γ′_{Θ,P′} (T (Θ, P′, δ))) .

Equations (5.63) and (5.62) imply

γ′_{Θ,P} (T (Θ, P, δ)) ∩ cl (γ′_{Θ,P′} (O_{1}^{asd}(Θ, P′, δ)))
⊂ γ′_{Θ,P} (T (Θ, P, δ)) ∩ cl (γ′_{Θ,P′} (T (Θ, P′, δ))) = ∅.

Thus, Lemma 5.16 implies there is a (smaller) SO(3) × SO(4) invariant neighborhood O_{2}^{asd}(ℝ^{4}, P) ⊂ O_{1}^{asd}(Θ, P, δ) of T (Θ, P, δ) with

γ′_{Θ,P} (O_{2}^{asd}(ℝ^{4}, P)) ∩ cl (γ′_{Θ,P} (O_{1}^{asd}(Θ, P, δ))) = ∅

as desired. Replace the open neighborhood O_{1}^{asd}(Θ, P, δ) with O_{2}^{asd}(ℝ^{4}, P). For all other P′′ ∈ [P], replace O_{1}^{asd}(Θ, P′′, δ) with the image of O_{2}^{asd}(ℝ^{4}, P) under the action of 𝔖_{κ}. This completes the case of P ≮ P′ and P′ ≮ P in the induction.

If [P] < P′, by conjugating P if necessary and using the invariance of the neighborhoods O_{1}^{asd}(Θ, P, δ) under the action of 𝔖_{κ}, we can assume P < P′. By Lemma 5.9, the closure of

[Θ] × (Σ(Z_{κ}(δ), P′) − Im(e(Z_{κ}, P))) ,

(where e(Z_{κ}, P) is the exponential map defined in (5.8)) does not intersect γ′_{Θ,P}(T (Θ, P, δ)). Thus, applying Lemma 5.16 and shrinking O_{1}^{asd}(Θ, P, δ), we can assume

cl (γ′_{Θ,P} (O_{1}^{asd}(Θ, P, δ))) ∩ ([Θ] × (Σ(Z_{κ}, P′′) − Im(e(Z_{κ}, P)))) = ∅

for all P′′ ∈ [P < P′] and thus

(5.64) cl (γ′_{Θ,P} (O_{1}^{asd}(Θ, P, δ))) ∩ ((∪_{P′′∈[P<P′]}γ′_{Θ,P′}(T (Θ, P′′, δ))) /𝔖(P))

⋐ [Θ] × e(Z_{κ}(δ), P) (O(Z_{κ}(δ), [P < P′])) ,

where O(Z_{κ}(δ), [P < P′]) is defined in Lemma 5.5. Let T (Θ, P, [P′], δ) and O_{1}^{asd}(Θ, P, [P′], δ) be the sets defined in (5.48). The equality and inclusion

[Θ] × e(Z_{κ}, P) (O(Z_{κ}(δ), [P < P′])) = γ′_{Θ,[P<P′]} (ρ_{P,[P′]}^{Θ,u} (T (Θ, P, [P′], δ)))
⊂ γ′_{Θ,[P<P′]} (ρ_{P,[P′]}^{Θ,u} (O_{1}^{asd}(Θ, P, [P′], δ))) ,

together with (5.64) then imply that

(5.65)  cl (γ′_{Θ,P} (O_{1}^{asd}(Θ, P, δ))) \ γ′_{Θ,P′} (ρ_{P,P′}^{Θ,u} (O_{1}^{asd}(Θ, P, P′, δ)))

is disjoint from γ′_{Θ,P′′}(T (Θ, P′′, δ)) for all P′′ ∈ [P < P′]. Lemma 5.16 gives an SO(3) × SO(4) × 𝔖(P) invariant neighborhood O_{1}^{asd}(Θ, [P < P′], δ) of

( ∐_{P′′∈[P<P′]} T (Θ, P′′, δ)) /𝔖(P) in ( ∐_{P′′∈[P<P′]} O^{asd}(Θ, P′′, δ)) /𝔖(P)

whose image under γ′_{Θ,[P<P′]} is disjoint from the last difference (5.65). This disjointess implies

γ′_{Θ,P} (O_{1}^{asd}(Θ, P, δ)) ∩ γ′_{Θ,P′′} (O_{1}^{asd}(Θ, P′′, δ))
⊂ γ′_{Θ,[P<P′]} (ρ_{P,[P′]}^{Θ,u} (O_{1}^{asd}(Θ, P, [P′], δ))) ,

for all P′′ ∈ [P < P′] and so as required in the conclusion of the lemma. □

Proof of Proposition 5.13. For each partition P of N_{κ}, let O_{1}^{asd}(Θ, P, δ) be the SO(3) × SO(4) invariant subspace of O^{asd}(Θ, P, δ) constructed in Lemma 5.17. For any two partitions P and P′ the images γ′_{Θ,P}(O_{1}^{asd}(Θ, P, δ)) and γ′_{Θ,P′}(O_{1}^{asd}(Θ, P′, δ)) are either disjoint or, by the last item in Lemma 5.17, are given by:

γ′_{Θ,P} ((ρ_{P,[P′]}^{Θ,d})^{−1}(O_{1}^{asd}(Θ, P, δ)) .

Thus, the overlaps are open subspaces and the union of the images W_{κ} has the desired properties. The estimates on F_{A}^{+} in the proposition follow from the arguments in [7], details will appear in [9, 10]. □

5.5. Tubular neighborhoods of the spliced end moduli space. We now introduce the tubular neighborhood structure of the spliced end W_{κ} defined in the previous section. This structure will satisfy the conditions discussed in the beginning of §4. In this lemma, we introduce the projection map. Lemma 5.18. For each partition P of N_{κ} with |P| > 1, let U(Θ, P) ⊂ W_{κ} be the image of the open subspace O_{1}^{asd}(Θ, P, δ) defined in Lemma 5.17. There is an SO(3) × SO(4) equivariant map

(5.66)  π(Θ, P) : U(Θ, P) → {[Θ]} × Σ(Z_{κ}(δ), P),

satisfying

(1) If P′ ∈ [P], then U(Θ, P) = U(Θ, P′), Σ(Z_{κ}(δ), P) = Σ(Z_{κ}(δ), P′), and π(Θ, P) =

π(Θ, P′).

(2) On the overlap U(Θ, P) ∩ U(Θ, P′),

(5.67)  π(Θ, P) ∘ π(Θ, P′) = π(Θ, P)

Proof. The map π(Θ, P) is defined to be the composition of the projection

O_{1}^{asd}(Θ, P, δ) → Σ(Z_{κ}, P),

with the inverse of the splicing map γ′_{Θ,P}. The equality (5.67) follows from (5.60). □

The obvious identification

(5.68)  {[Θ]} × Z_{κ}/𝔖_{κ} ≅ Z_{κ}/𝔖_{κ},

allows us to define the projection maps π(Z_{κ}, P) and functions t(Z_{κ}) defined in §4.1 as maps and functions on the trivial strata of W_{κ}. We identify the restriction of the projection π(Θ, P) with such a map in the following. Lemma 5.19. Under the identification (5.68), the restriction of the projection map π(Θ, P) of Lemma 5.18 to the intersection

({[Θ]} × Z_{κ}/𝔖_{κ}) ∩ U(Θ, P)

is equal to the map π(Z_{κ}, P) : U(Z_{κ}, P)/𝔖(P) → Δ^{∘}(Z_{κ}, P) = Σ(Z_{κ}, P)/𝔖(P) defined in (5.67).

Proof. The lemma follows from noting that the restriction of the splicing map γ′_{Θ,P} to points where the connections being spliced in are trivial is exactly the map e(Z_{κ}, P) of (4.9). □

Before defining the tubular distance functions, we define a perturbation of the scale function which is constant along the fibers of the projections π(Θ, P). This scale shall be used in the construction of the tubular distance function and in the definition of the link of the reducible PU(2) monopoles. Lemma 5.20. There are continuous maps

(5.69)  λ̃_{κ} : M̄_{spl,κ}^{s,♮}(δ) → [0, 2δ),

and

(5.70)  t_{r}(Θ, P) : U(Θ, P) → [0, 1], and t(Θ, P) : U(Θ, P) → [0, 1].

which are smooth on each stratum, SO(3) × SO(4) invariant, and satisfy for some neighborhoods U′(Θ, P) ⋐ U(Θ, P) of {[Θ]} × Σ(Z_{κ}(δ), P)

(1) The function λ̃_{κ} is equal to the scale function λ on the complement of W_{κ},

(2) Under the identification (5.68), the restriction of λ̃_{κ} to the trivial strata in (5.68) is equal to the function t(Z_{κ}) defined in Lemma 4.3,

(3) For all partitions P of N_{κ} with |P| > 1, λ̃_{κ} = λ̃_{κ} ∘ π(Θ, P) on U′(Θ, P),

(4) If P′ ∈ [P] then, t_{r}(Θ, P) = t(Θ, P′) and t(Θ, P) = t̃(Θ, P′),

(5) On the overlap U′(Θ, P) ∩ U′(Θ, P′),

(5.71)  t_{r}(Θ, P) ∘ π(Θ, P′) = t_{r}(Θ, P), and t(Θ, P) ∘ π(Θ, P′) = t(Θ, P)

(6) t_{r}(Θ, P)^{−1}(0) = t(Θ, P)^{−1}(0) = {Θ} × Σ(Z_{κ}(δ), P),

(7) There are open neighborhoods

U_{1}(Θ, P) ⋐ U_{2}(Θ, P) ⋐ U(Θ, P)

of {Θ} × Σ(Z_{κ}, P), such that

(5.72)  \begin{aligned}U_{1}(Θ, P) & = t_{r}(Θ, P)^{−1}([0, \frac{1}{2})), \\ U(Θ, P) − U_{2}(Θ, P) & = t_{r}(Θ, P)^{−1}(\{1\}) \end{aligned}

Proof. The function λ̃_{κ} is constructed using induction on κ. Thus, we assume the functions λ̃_{κ′}, t_{r}(Θ, P), and t(Θ, P) have been constructed for all κ′ < κ and all partitions P of N_{κ′}.

Begin the induction with λ̃_{1} = λ_{1}. First, observe that the restriction of λ̃_{κ} to the trivial strata in W_{κ} appearing in (5.68) is determined by item (2). This is consistent with item (3) by virtue of the identification of π(Θ, P) in Lemma 5.19 and the equality in item (2) of Lemma 4.3. Then construct λ̃_{κ} following the pattern of the construction of the function t(Z_{κ}) in Lemma 4.3 using t_{r}(Θ, P) in place of t_{r}(Z_{P} , P).

For each partition P of N_{κ}, the function t(Θ, P) is then defined by

(5.73)  t(Θ, P) = ∑_{P ∈P} |P |λ̃_{P}^{2} ,

where the functions λ̃_{P} are defined on O^{asd}(Θ, P, δ) by the new scale function λ̃_{|P |} on the S^{4} connection being spliced in at x_{P}.

The equality (5.67) holds for t(Θ, P) by the same argument used to establish (4.38), using the observation that for P < P′, the projection π(Θ, P′) on U(Θ, P) ∩ U(Θ, P′) is given by the restriction of the projection

Δ^{∘}(Z_{κ}(δ), P) × ∏_{P ∈P} (Δ^{∘}(Z_{|P |}(δ_{P}), P′_{P}) × ∏_{P′∈P′_{P}} M̄_{spl,|P′|}^{s,♮}(δ_{P′}))
↓
Δ^{∘}(Z_{κ}(δ), P) × ∏_{P ∈P} (Δ^{∘}(Z_{|P |}(δ_{P}), P′_{P}) × ∏_{P′∈P′_{P}} {[Θ, c_{P′}]})

to the appropriate subspaces (see (5.52) and (5.55)).

The function t_{r}(Θ, P) is then a rescaling of the function t(Θ, P) in the sense of Lemma 4.12. That is, t_{r}(Θ, P) is defined by multiplying t(Θ, P) by a non-vanishing, smooth function pulled back from T (Θ, P, δ) by the map π(Θ, P). The property (5.67) of π(Θ, P) implies that the pullback of a function f by π(Θ, P) satisfies

(f ∘ π(Θ, P)) ∘ π(Θ, P′) = f ∘ π(Θ, P)

so defining t_{r}(Θ, P) by such a rescaling of t(Θ, P) implies that if t(Θ, P) has the property (5.67) then t_{r}(Θ, P) does as well. □

5.6. The isotopy of the spliced end. The final step in the proof of Theorem 5.1 is to use the isotopy provided by the gluing and centering maps to piece together the spliced end W_{κ} with the actual moduli space M̄_{κ}^{s,♮}(δ).

We first construct the isotopy of W_{κ} in B̄_{κ}^{s}(δ). Lemma 5.21. There is continuous, smoothly-stratified map

(5.74)  R : (−∞, 1] × W_{κ} → B̄_{κ}^{s}(δ),

such that (1) For all t ∈ (−∞, \frac{1}{2}] and all [A, F^{s}, x] ∈ W_{κ}, R(t, [A, F^{s}, x]) = [A, F^{s}, x],

(2) For all t ∈ [\frac{3}{4}, 1] and all [A, F^{s}, x] ∈ W_{κ}, R(t, [A, F^{s}, x]) ∈ M̄_{κ}^{s}(δ),

(3) For all [A, F^{s}, x] ∈ W_{κ}, R(1, [A, F^{s}, x]) ∈ M̄_{κ}^{s,♮}(δ),

(4) For all t ∈ (−∞, 1], the map R(t, ·) : W_{κ} → B̄_{κ}^{s}(δ) is SO(3) × SO(4) equivariant.

Proof. The bound on the L^{♯,2}-norm of the F_{A}^{+} for all [A, x] ∈ W_{κ} implies (see [7, Proposition 7.6]) that there is a continuous, smoothly-stratified, SO(3) × SO(4) equivariant embedding, G : W_{κ} → M̄^{s,♮}(δ)

(5.75)  G([A, F^{s}, x]) = [A + a(A), F^{s}, x].

For t ∈ [\frac{1}{2}, \frac{3}{4}], we define

(5.76)  R(t, [A, F^{s}, x]) = [A + 4(t − \frac{1}{2})a(A), F^{s}, x].

The remainder of the isotopy, for t ∈ [\frac{3}{4}, 1] is defined by the mass-centering map. That is, pulling any [A, F^{s}, x] back by an appropriate conformal diffeomorphism ensures it is mass-centered (see [6, §3.2.1]). This defines an natural isotopy, parameterized by t ∈ [\frac{3}{4}, 1], of the image of W_{κ} under R(\frac{1}{2}, ·) into the space of mass-centered connections. □

Proof of Theorem 5.1. The function 1 − t_{r}(Θ, P) is compactly supported on U(Θ, P) and thus defines an SO(3) × SO(4)-invariant function on W_{κ}. Observe that by (5.72), the map

(5.77)  t = 1 − ∑_{P}(1 − t_{r}(Θ, P)).

satisfies

t(∪_{P}U_{1}(Θ, P)) ⊂ (−∞, \frac{1}{2}],
t (∩_{P}(W_{κ} − U_{2}(Θ, P))) = 1.

That is, if [A, F^{s}, x] ∈ U_{1}(Θ, P) for some P then t([A, F^{s}, x]) < \frac{1}{2} while if [A, F^{s}, x] ∉ U_{2}(Θ, P) for all P, then t([A, F^{s}, x]) = 1. Alternately, t < \frac{1}{2} on the “interior” of W_{κ}, near the trivial strata while t is equal to one the “exterior”, away from the trivial strata. Define R(W_{κ}) to be the image of W_{κ} under the map R(t(·), ·):

R(W_{κ}) = {R(t([A, F^{s}, x]), [A, F^{s}, x]) : [A, F^{s}, x] ∈ W_{κ}}.

Then if we define the spliced end moduli space by

M̄_{spl,κ}^{s,♮}(δ) = (M̄_{κ}^{s,♮}(δ) − R(1, W_{κ})) ∪ R(W_{κ}),

it will have the properties specified in Theorem 5.1. □

6. The space of global splicing data

In this section, we construct the space of global splicing data. This space will be the union of spaces of splicing data, O(𝔱, P), associated to diagonals Σ(X^{ℓ}, P), as P varies over partitions of N_{ℓ}. (Of course, we will do this equivariantly with respect to the symmetric group.) To each partition P, we will define a crude splicing map which is identical to the splicing map defined in [9] except

(1) The connections on S^{4} being spliced in are elements of the spliced ends moduli space rather than the moduli space of anti-self-dual connections on S^{4},

(2) The background pair, (A_{0}, Φ_{0}), are “flattened” (i.e. the connection A_{0} is replaced with one which is flat while the section Φ_{0} is multiplied by a vanishing cut-off function) on balls of fixed radius around the splicing point rather than balls whose radius vanishes as the connection on S^{4} becomes more concentrated.

This last difference is necessary to mimic the result which holds when splicing connections to the trivial connection on ℝ^{4} (Proposition 5.12), allowing us to control the overlap of the images of the crude splicing map as we did when constructing the spliced end W_{κ} by defining a space of overlap data and overlap maps. With this control of the overlaps of the images of the crude splicing maps, we can define the space of global splicing data is then the union of the images of the crude splicing maps.

After constructing the domain of the splicing map in §6.1, we show how to flatten pairs as described above in §6.2 which defines the crude splicing map in §6.3. With the understanding of the overlaps of the images of the crude splicing maps developed in 6.4, we construct the space of global splicing data in §6.5. In §6.6, we construct the partial Thom-Mather structure which will be used in the construction of the link L̄_{𝔱,𝔰}. In §6.7, we produce a global splicing map whose image will contain the reducible monopoles. Finally, in §6.8, we construct a projection map from the space of global splicing data to Sym^{ℓ}(X).

6.1. The splicing data. We begin by recalling the domains of the splicing maps from [9].

6.1.1. The background pairs. Let M̃_{𝔰} ⊂ C̃_{𝔱_{0}} be the image of the embedding of the solutions of the perturbation of the Seiberg-Witten equations considered in [15]. The gauge group quotient, M_{𝔰} = M̃_{𝔰}/G_{𝔰} embeds in C_{𝔱_{0}}. A virtual normal bundle N_{𝔱_{0},𝔰} → M_{𝔰} with an embedding of a δ-disk subbundle N_{𝔱_{0},𝔰}(δ) ⊂ N_{𝔱_{0},𝔰} → C_{𝔱_{0}}, was defined in [15]. We will abbreviate N_{𝔱_{0},𝔰}(δ) by N(δ). If Ñ → M̃_{𝔰} is the pullback of N_{𝔱_{0},𝔰} by the projection M̃_{𝔰} → M_{𝔰}, we showed in [11] that there is an embedding Ñ(δ) → C̃_{𝔱_{0}} covering the embedding of N(δ) → C_{𝔱_{0}}.

6.1.2. The metrics. For each partition P of N_{ℓ}, let g_{P} be the smooth family of metrics on X parameterized by Δ^{∘}(X^{ℓ}, P) and let Ũ(X^{ℓ}, P) ⊂ X^{ℓ} be the tubular neighborhood of Δ^{∘}(X^{ℓ}, P) constructed in Lemma 4.11. Recall that π(X^{ℓ}, P) : Ũ(X^{ℓ}, P) → Δ^{∘}(X^{ℓ}, P) is the projection map of this tubular neighborhoods. We list here the relevant properties:

(1) For all y ∈ Δ^{∘}(X^{ℓ}, P), the metric g_{P,y} is flat on the support of Ũ(X^{ℓ}, P)∩π(X^{ℓ}, P)^{−1}(y),

(2) For P < sP′ partitions of N_{ℓ} and for y′ ∈ Ũ(X^{ℓ}, P)∩Δ^{∘}(X^{ℓ}, P′) with π(X^{ℓ}, P)(y′) = y, the equality g_{P,y} = g_{P,y′} holds,

(3) The metrics are 𝔖_{ℓ} invariant in the sense that g_{P,y} = g_{σ(P),σ(y)} for all σ ∈ 𝔖_{ℓ}.

6.1.3. The frame bundles. For P be a partition of N_{ℓ}, let

Fr(T X^{ℓ}, P, g_{P}) → Δ^{∘}(X^{ℓ}, P),

be the fiber bundle defined before (4.18). Similarly, for 𝔤_{0} = 𝔤_{V_{0}} we define

Fr(𝔤_{0}, P) → Δ^{∘}(X^{ℓ}, P),

as

(6.1) Fr(𝔤_{0}, P) = {(F_{1}^{𝔤}, …, F_{ℓ}^{𝔤}) ∈ ∏_{i=1}^{ℓ} Fr(𝔤_{0}) :

F_{i}^{𝔤} = F_{j}^{𝔤} if and only if there is P ∈ P and i, j ∈ P }.

Then, the gluing data bundle is:

(6.2)  Fr(𝔱, 𝔰, P, g_{P}) = Fr(T X, P, g_{P}) ×_{Δ^{∘}(X^{ℓ},P)} Fr(𝔤_{𝔱_{0}}, P).

Recall that we denote elements of Δ^{∘}(X^{ℓ}, P) as (y_{P})_{P ∈P} where y_{P} ∈ X. We will similarly denote elements of the bundle Fr(𝔱, 𝔰, P, g_{P}) lying over (y_{P})_{P ∈P} as (F_{P}^{T} , F_{P}^{𝔤})_{P ∈P} where F_{P}^{T} ∈ Fr(T X)|_{y_{P}} and F_{P}^{𝔤} ∈ Fr(𝔤_{𝔱_{0}})|_{y_{P}}. We also write (F_{P}^{T,𝔤})_{P ∈P} for such an element of Fr(𝔱, 𝔰, P, g_{P}).

The structure groups of Fr(X, P) over Δ^{∘}(X^{ℓ}, P) and over Σ(X^{ℓ}, P) are G̃(P) and G(P) respectively where

(6.3)

G̃(P) = {((R_{1}, M_{1}), …, (R_{ℓ}, M_{ℓ})) ∈ (SO(4) × SO(3))^{ℓ} :
for every i, j ∈ P ∈ P R_{i} = R_{j} and M_{i} = M_{j}},
G(P) = G̃(P) ⋉ 𝔖(P).

A GENERAL FORMULA RELATING DONALDSON AND SEIBERG-WITTEN INVARIANTS 49

Observe that for P < P′, there is an inclusion,

(6.4) G̃(P) → G̃(P′).

We can write elements of G̃(P) as (R_{P} , M_{P})_{P ∈P}. With this notation the action of G(P) on

∏_{P ∈P} M̄_{spl,|P |}^{s,♮}(δ)

is given by the factor (R_{P} , M_{P}) acting by the standard SO(3) × SO(4) action on M̄_{spl,|P |}^{s,♮}(δ) and the natural permutation action of 𝔖(P). 6.1.4. The space of splicing data. The splicing map will be defined on a subspace

Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P)

where

(6.5)  O(𝔱, 𝔰, P) ⊂ Gl(𝔱, 𝔰, P, g_{P}) = Fr(𝔱, 𝔰, P, g_{P}) ×_{G(P)} ∏_{P ∈P} M̄_{spl,|P |}^{s,♮}(δ_{P}),

is an open neighborhood of

(6.6)  Σ(𝔱, 𝔰, P, g_{P}) = {((F_{P}^{T,𝔤})_{P ∈P}, ([Θ, F_{P}^{s} , c_{|P |}])_{P ∈P}) ∈ Gl(𝔱, 𝔰, P, g_{P})}

where c_{|P |} is the cone point of Sym^{|P |,♮}(ℝ^{4}). The notation in (6.6) is motivated by the observation that because SO(3) × SO(4) acts trivially on [Θ, F_{P}^{s} , c_{|P |}], there is an identification,

(6.7)  Σ(𝔱, 𝔰, P, g_{P}) ≅ Σ(X^{ℓ}, P).

We also define a larger subspace of Gl(𝔱, 𝔰, P, g_{P}) by:

(6.8)  T (𝔱, 𝔰, P, g_{P}) = {((F_{P}^{T,𝔤})_{P ∈P}, ([Θ, F_{P}^{s} , v_{P}])_{P ∈P}) ∈ Gl(𝔱, 𝔰, P, g_{P})}.

One requirement in the definition of O(𝔱, 𝔰, P) will be that for any point,

A = ((A_{0}, Φ_{0})(F_{P}^{T} , F_{P}^{𝔤})_{P ∈P}, ([A_{P} , F_{P}^{s} , v_{P}])_{P ∈P}) ∈ O(𝔱, 𝔰, P)

where F_{P}^{T} and F_{P}^{𝔤} lie over y_{P} ∈ X that

(6.9)  8λ([A_{P} , F_{P}^{s} , v_{P}]) < min_{P′∈P,P′≠P} d(y_{P} , y′_{P})

for all P ∈ P. Finally, we define

(6.10)  π_{Σ} : Ñ(δ) ×_{G_{𝔰}} Gl(𝔱, 𝔰, P, g_{P}) → Σ(X^{ℓ}, P)

to be the obvious projection. The S^{1} action on Ñ_{𝔱_{0},𝔰} defined in 2 defines an S^{1} action on

Ñ(δ) ×_{G_{𝔰}} Gl(𝔱, 𝔰, P, g_{P})

and we will see that all the constructions of this section are equivariant with respect to this S^{1} action. See [11] for different ways of presenting this S^{1} action.

6.2. Flattening pairs. The commutativity of the diagram (5.52) for the spliced ends moduli space depended on the flatness of the metric on ℝ^{4} and on the flatness of the trivial connection Θ. The locally flat metrics g_{P} described in §6.1.2 have this property. We now introduce a method for achieving the same kind of local flatness for the background connection A_{0}. This method also yields a way of cutting off the background section consistently from one stratum to another. Definition 6.1. A collection of smooth functions s_{P} : Δ^{∘}(X^{ℓ}, P) → (0, 1), indexed by P ∈ P are a separating family if

(1) The functions are 𝔖(P) invariant in the sense that s_{P} = s_{σ(P)} and s_{P} = s_{P} ∘ σ, for

all σ ∈ 𝔖(P),

(2) For all y = (y_{P})_{P ∈P} ∈ Δ^{∘}(X^{ℓ}, P) and for all P ≠ P′ ∈ P, the balls B(y_{P} , 4s_{P} (y)^{1/2})

and B(y_{P′}, 4s_{P′}(y)^{1/2}) are disjoint.

Definition 6.2. Let {s_{P} } be a separating family of smooth functions on Δ^{∘}(X^{ℓ}, P). A connection Â_{0} is flat with respect to {s_{P} } at y = (y_{P})_{P ∈P} ∈ Δ^{∘}(X^{ℓ}, P) if the connection Â_{0} is flat on

(6.11)  ∪_{P ∈P} B(y_{P} , 2s_{P} (y)^{1/2}).

A pair (A, Φ) ∈ C_{𝔱_{0}} is flat with respect to {s_{P} } if Â is flat with respect to {s_{P} } and Φ vanishes on

(6.12)  ∪_{P ∈P} B(y_{P} , 4s_{P} (y)^{1/3}).

The following lemma is the analogue of the flattening construction for metrics given in Lemma 4.10. Lemma 6.3. Let P be a partition of N_{ℓ}. Let {s_{P} } be a separating family of smooth functions on Δ^{∘}(X^{ℓ}, P). Then, there is an S^{1}-equivariant map,

(6.13)  Θ′_{P} : Ñ(δ) × Δ^{∘}(X^{ℓ}, P) → C̃_{𝔱_{0}}

which is G_{𝔰}-equivariant and descends to the quotient

N(δ) × Δ^{∘}(X^{ℓ}, P) → C_{𝔱_{0}}

In addition, for all y = (y_{P})_{P ∈P} ∈ Δ^{∘}(X^{ℓ}, P), if (A′, Φ′) = Θ′_{P}((A_{0}, Φ_{0}), y) then

(1) The connection Â′ is flat on

∪_{P ∈P}B(y_{P} , 2s_{P} (y)^{1/2}),

and equal to A_{0} on the complement of

(6.14)  ∪_{P ∈P}B(y_{P} , 4s_{P} (y)^{1/2})

(2) The section Φ′ vanishes on

∪_{P ∈P}B(y_{P} , 4s_{P} (y)^{1/3})

and is equal to Φ on the complement of

(6.15)  ∪_{P ∈P}B(y_{P} , 8s_{P} (y)^{1/3}).

If the connection Â_{0} is already flat on the space (6.14), then A′ = A_{0}. If the section Φ already vanishes on the space (6.15), then Φ′ = Φ_{0}.

Proof. The map Θ_{P′} is initially defined as a map,

Θ′_{P} : Ñ(δ) × Fr(𝔱, 𝔰, P, g_{P}) → C̃_{𝔱_{0}}

as described in [9, §3.2]. Specifically, the frames and radial parallel translation define a trivial connection on the space (6.14). If Â_{0} is flat on the space (6.14), this trivial connection is equal to Â_{0}. Then, use a cut-off function to interpolate between the trivial connection and Â_{0} on the annuli around the points y_{P}, defining a new connection Â′ which is flat on the inner balls and equal to Â_{0} outside the space (6.14). Observe that the resulting connection Â′ is independent of the choice of the frames used as it is equal to the connection obtained by splicing in the trivial connection which has stabilizer SO(3) × SO(4). Thus, the map Θ′_{P} descends to the domain stated in the lemma.

The section Φ′ is defined by multiplying Φ_{0} by a cut-off function equal to on the space (6.15) and vanishing on the balls of radius 4s_{P} (y)^{1/3}. □

To define the crude splicing maps in such a way that an analogue of Proposition 5.12 controls the overlaps of their images, we must redefine the flattening maps in such a way that for y′ ∈ Δ^{∘}(X^{ℓ}, P′) near y ∈ Δ^{∘}(X^{ℓ}, P) the flattened background pairs Θ_{P′}(A_{0}, Φ_{0}, y′) = Θ_{P}(A_{0}, Φ_{0}, y). (Compare the second property of the metrics g_{P} appearing in §6.1.2.) To this end, we introduce a refinement of the flattening map of Lemma 6.3. Lemma 6.4. Continue the assumptions of Lemma 6.3. Then, for every partition P of N_{ℓ} there are separating functions {s̃_{P} } and is an S^{1}-equivariant map

(6.16)  Θ_{P} : Ñ(δ) × Δ^{∘}(X^{ℓ}, P) → C̃_{𝔱_{0}},

which is 𝔖(P)-invariant, G_{𝔰}-equivariant and descends to the quotient

N(δ) × Δ^{∘}(X^{ℓ}, P) → C_{𝔱_{0}}

such that if (A′, Φ′) = Θ_{P}(A_{0}, Φ_{0}, y) where y = (y_{P})_{P ∈P} ∈ Δ^{∘}(X^{ℓ}, P) then (1) The connection Â′ is flat on

∪_{P ∈P}B(y_{P} , 2s̃_{P} (y)^{1/2}),

(2) The section Φ′ vanishes on

∪_{P ∈P}B(y_{P} , 4s̃_{P} (y)^{1/3}).

Furthermore, if P < P′ and the tubular neighborhood U(X^{ℓ}, P) is suitably small, if y′ ∈ U(X^{ℓ}, P) ∩ Δ^{∘}(X^{ℓ}, P′) and π(X^{ℓ}, P)(y′) = y ∈ Δ^{∘}(X^{ℓ}, P), then

(6.17)  Θ_{P}(A_{0}, Φ_{0}, y) = Θ_{P′}(A_{0}, Φ_{0}, y′).

Proof. The proof is identical to the construction of the consistent flat families of metrics in Lemma 4.11. One constructs Θ_{P} by induction on P.

For P_{0} = {N_{ℓ}} the crudest partition of N_{ℓ}, define Θ_{P_{0}} = Θ′_{P_{0}} (as constructed in Lemma 6.3).

Assume Θ_{P} has been constructed for all P < P′. To construct Θ_{P′}, we first define

Θ_{P,P′} : Ñ(δ) × (U(X^{ℓ}, P ∩ Δ^{∘}(X^{ℓ}, P′)) → C̃_{𝔱_{0}},

by

Θ_{P′,P}(A_{0}, Φ_{0}, y′) = Θ_{P}(A_{0}, Φ_{0}, π(X^{ℓ}, P)(y′)).

Observe that if P_{1} < P and P_{2} < P and U(X^{ℓ}, P_{1}) ∩ U(X^{ℓ}, P_{2}) is non-empty, then we can assume P_{1} < P_{2} < P. The Thom-Mather condition on the projections π(X^{ℓ}, P), (4.30):

π(X^{ℓ}, P_{1}) ∘ π(X^{ℓ}, P_{2}) = π(X^{ℓ}, P_{1}),

and the inductive assumption imply that for y′ ∈ U(X^{ℓ}, P_{1}) ∩ U(X^{ℓ}, P_{2}) ∩ Δ^{∘}(X^{ℓ}, P′),

Θ_{P′,P_{2}}(A_{0}, Φ_{0}, y′) = Θ_{P_{2}}(A_{0}, Φ_{0}, π(X^{ℓ}, P_{2})(y′))

= Θ_{P_{1}}(A_{0}, Φ_{0}, π(X^{ℓ}, P_{1})π(X^{ℓ}, P_{2})(y′)) By inductive hypothesis

= Θ_{P_{1}}(A_{0}, Φ_{0}, π(X^{ℓ}, P_{1})(y′)) By Thom-Mather property (4.30)

= Θ_{P′,P_{1}}(A_{0}, Φ_{0}, y′).

By the preceding equality, we can define

Θ_{P′,U}(A_{0}, Φ_{0}, y′) = Θ_{P′,P}(A_{0}, Φ_{0}, y′) for y′ ∈ U(X^{ℓ}, P)

on Ñ(δ) × U(P′) where

(6.18)  U(P′) = ∪_{P<P′}U(X^{ℓ}, P) ∩ Δ^{∘}(X^{ℓ}, P′).

Then, we define

Θ_{P′,1} : Ñ(δ) × Δ^{∘}(X^{ℓ}, P′) → C̃_{𝔱_{0}},

by

Θ_{P′,1}(A_{0}, Φ_{0}, y′) = (1 − χ(y′))(A_{0}, Φ_{0}) + χ(y′)Θ_{P′,U}(A_{0}, Φ_{0}, y′)

where χ : Δ^{∘}(X^{ℓ}, P) → [0, 1] is an 𝔖(P) invariant function supported in (6.18) and equal to one on a neighborhood U_{2} of the end of Δ^{∘}(X^{ℓ}, P) (hence U_{2} is a proper subspace of U(P′)). By shrinking the tubular neighborhoods U(X^{ℓ}, P), we can find a family of separating functions {s̃_{P′}}_{P′∈P′}, such that for y′ = (y′_{P′})_{P′∈P′} ∈ U(X^{ℓ}, P) ∩ Δ^{∘}(X^{ℓ}, P′) with π(X^{ℓ}, P)(y′) = y = (y_{P})_{P ∈P}

(6.19)  \begin{matrix}B(y′_{P′}, 8s̃_{P′}(y′)^{1/3}) ⋐ B(y_{P} , 4s̃_{P} (y)^{1/3}) \\ and B(y′_{P′}, 4s̃_{P′}(y′)^{1/2}) ⋐ B(y_{P} , 2s̃_{P} (y)^{1/2}) \end{matrix}

for all P′ ∈ P′_{P}. By these inclusions, by the inductive assumption, and by the final statement of Lemma 6.3, we have the equality, for y′ ∈ U_{2},

Θ′_{P′} (Θ_{P,1}(A_{0}, Φ_{0}, y′), y′) = Θ_{P,1}(A_{0}, Φ_{0}, y′).

That is, for y′ ∈ U_{2}, the pair Θ_{P,1}(A_{0}, Φ_{0}, y′) is already flat on the relevant balls and thus the flattening construction of Lemma 6.3 does not change the pair. Hence, if we define

Θ_{P′}(A_{0}, Φ_{0}, y′) = Θ′_{P′}(Θ_{P,1}(A_{0}, Φ_{0}, y′), y′),

this flattening map will satisfy (6.17), completing the construction of the map Θ_{P′} and thus completing the induction. □

6.3. The crude splicing map. The crude splicing map:

(6.20)  γ′′_{𝔱,𝔰,P} : Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P}) → C̄_{𝔱}

is S^{1}-equivariant and defined on the domain

(6.21) \begin{matrix}O(𝔱, 𝔰, P, g_{P}) \\ = \{(F_{P}^{T,𝔤}, [A_{P} , F_{P}^{s} , v_{P}])_{P ∈P} ∈ Gl(𝔱, 𝔰, P, g_{P}) : 8λ([A_{P} , F_{P}^{s} , v_{P}])^{1/3} ≤ s_{P} (F_{P}^{T,𝔤})\}, \end{matrix}

where {s_{P} } is the pullback of the family of separating functions {s̃_{P} } constructed in Lemma 6.4 by the projection Gl(𝔱, 𝔰, P, g_{P}) → Σ(X^{ℓ}, P).

The crude splicing map is defined exactly as is the splicing map in [9, §3.2] with the following exceptions.

First, the connections on S^{4} being spliced in are elements of the spliced ends moduli space M̄_{spl,|P |}^{s,♮} constructed in §5 rather than elements of the anti-self-dual moduli space M̄_{|P |}^{s,♮}.

Second, the metric on X used to identify balls in X with balls around the north pole of S^{4} is fixed for the standard splicing map. In the crude splicing map, the metric varies with the splicing points y ∈ Δ^{∘}(X^{ℓ}, P), varying in a smooth family of metrics g_{P}.

Third, in the definition of the splicing map in [9, §3.2], the background connection A_{0} is flattened on the balls

B(y_{P} , 2λ_{P}^{1/2})

while the background section is multiplied by a cut-off function vanishing on

B(y_{P} , 4λ_{P}^{1/3}),

where λ_{P} is the scale of the connection on S^{4} being spliced in at x_{P} ∈ X. The crude splicing map is defined instead by replacing the background pair (A_{0}, Φ_{0}) with the background pair Θ_{P}(A_{0}, Φ_{0}, y) constructed in Lemma 6.4. Thus, the background pair is flattened on balls whose radius does not depend on the scale of the connections on S^{4}.

The definition of the crude splicing map immediately implies the following observation about the crude splicing map. Lemma 6.5. Let Σ(𝔱, 𝔰, P, g_{P}) be as defined in (6.6). Let id_{Σ} : Σ(𝔱, 𝔰, P, g_{P}) ≅ Σ(X^{ℓ}, P) be the identification in (6.7). Then the restriction of the crude splicing map γ′′_{𝔱,𝔰,P} to

Ñ(δ) ×_{G_{𝔰}} Σ(𝔱, 𝔰, P, g_{P}) = N(δ) × Σ(𝔱, 𝔰, P, g_{P})

takes values in

C_{𝔱_{0}} × Σ(X^{ℓ}, P) ⊂ C̄_{𝔱}

and for any F ∈ Σ(𝔱, 𝔰, P, g_{P}) and (A_{0}, Φ_{0}) ∈ Ñ_{𝔱_{0},𝔰},

γ′′_{𝔱,𝔰,P}((A_{0}, Φ_{0}), F) = (Θ_{P}(A_{0}, Φ_{0}, id_{Σ}(F)), id_{Σ}(F)) .

Remark 6.6. In contrast to the result of Lemma 6.5, the standard splicing map would satisfy

γ′_{X,P}((A_{0}, Φ_{0}), F) = ([A_{0}, Φ_{0}], id_{Σ}(F)) ,

that is, the background pair are not flattened.

6.4. The overlap spaces. We now introduce spaces of overlap data Gl(𝔱, 𝔰, P, [P′]), attempting to mimic the construction of the spliced end in §5.

6.4.1. The overlap space. For P < P′ partitions of N_{ℓ} and g_{P} a family of smooth metrics on X parameterized by Δ^{∘}(X^{ℓ}, P), we define the space of overlap data by:

(6.22) \begin{matrix}Gl(𝔱, 𝔰, P, [P′], g_{P}) \\ = Fr(𝔱, 𝔰, P, g_{P}) ×_{G(P)} ∐_{P′′∈[P<P′]} ∏_{P ∈P} (Δ^{∘}(Z_{|P |}(δ_{P}), P′′_{P}) × ∏_{Q∈P′′_{P}} M̄_{spl,|Q|}^{s,♮}(δ_{Q})) . \end{matrix}

Recall that we denote elements of Δ^{∘}(Z_{|P |}(δ_{P}), P′′_{P}) as (v_{Q})_{Q∈P′′_{P}} where v_{Q} ∈ ℝ^{4}. We then denote elements of Gl(𝔱, 𝔰, P, [P′], g_{P}) by

(6.23) A = ((F_{P}^{T,𝔤})_{P ∈P}, ((v_{Q})_{Q∈P′′_{P}} , ([A_{Q}, F_{Q}^{s}, v_{Q}])_{Q∈P′′_{P}} )_{P ∈P}) ∈ Gl(𝔱, 𝔰, P, [P′], g_{P}),

where (F_{P}^{T,𝔤})_{P ∈P} ∈ Fr(𝔱, 𝔰, P, g_{P}), (v_{Q})_{Q∈P′′_{P}} ∈ Δ^{∘}(Z_{|P |}, P′′_{P}), and [A_{Q}, F_{Q}^{s}, v_{Q}] ∈ M̄_{spl,|Q|}^{s,♮}(δ). The projection Fr(𝔱, 𝔰, P, g_{P}) → Σ(X^{ℓ}, P) defines a projection,

(6.24)  π_{Σ} : Gl(𝔱, 𝔰, P, [P′], g_{P}) → Σ(X^{ℓ}, P).

Because [Θ, F^{s}, c_{|Q|}] ∈ M̄_{spl,|Q|}^{s,♮} is a fixed point of the action of SO(3) × SO(4), the subspace

(6.25) \begin{matrix}Σ(𝔱, 𝔰, P, [P′], g_{P}) \\ = \{((F_{P}^{T,𝔤})_{P ∈P}, ((v_{Q})_{Q∈P′′_{P}} , ([Θ, F_{Q}^{s}, c_{|Q|}])_{Q∈P′′_{P}} )_{P ∈P}) ∈ Gl(𝔱, 𝔰, P, [P′], g_{P})\} \end{matrix}

is identified with the normal bundle ν(X^{ℓ}, P, [P′])

(6.26)  k_{Σ,P,[P′]} : Σ(𝔱, 𝔰, P, [P′], g_{P}) ≅ ν(X^{ℓ}, P, [P′])

as, the reader may recall, ν(X^{ℓ}, P, [P′]) is defined by

Fr(T X^{ℓ}, P, g_{P}) ×_{G_{T} (P)} ∐_{P′′∈[P<P′]} Δ^{∘}(Z_{|P |}(δ_{P}), P′′_{P}).

There is thus a projection,

(6.27)  π_{Σ,P,[P′]} : Gl(𝔱, 𝔰, P, [P′], g_{P}) → Σ(𝔱, 𝔰, P, [P′], g_{P}),

given by

π_{T} ((F_{P}^{T} , F_{P}^{𝔤})_{P ∈P}, ((v_{Q})_{Q∈P′′_{P}} , ([A_{Q}, F_{Q}^{s}, x_{Q}])_{Q∈P′′_{P}} )_{P ∈P})
= ((F_{P}^{T} , F_{P}^{𝔤})_{P ∈P}, ((v_{Q})_{Q∈P′′_{P}} , ([Θ, F_{Q}^{s}, c_{|Q|}])_{Q∈P′′_{P}} )_{P ∈P}) .

We will then write

(6.28)  π_{ν(Σ)} : Gl(𝔱, 𝔰, P, [P′], g_{P}) → ν(X^{ℓ}, P, [P′]), π_{ν(Σ)} = k_{Σ,P,[P′]} ∘ π_{Σ,P,[P′]},

for the projection to the normal bundle.

6.4.2. The upwards overlap map. To work simultaneously with all the diagonals P′′ ∈ [P < P′], we introduce the gluing space,

Gl(𝔱, 𝔰, [P < P′], g_{P′}) = ( ∐_{P′′∈[P<P′],g_{P′′}} Fr(𝔱, 𝔰, P′′, g_{P′′}) ×_{G̃(P′′)} ∏_{Q∈P′′} M̄_{spl,|Q|}^{s,♮}) /𝔖(P).

Note that by the 𝔖(P) invariance of the families of metrics g_{P′′} (see §6.1.2), the space Gl(𝔱, 𝔰, [P < P′], g_{P′}) depends only on the choice of one family of metrics g_{P′}. By comparing the orbits under the symmetric group actions, one can see there is an identification

Gl(𝔱, 𝔰, P′′, g_{P′′}) ≃ Gl(𝔱, 𝔰, [P < P′], g_{P′}),

for any P′′ ∈ [P < P′]. Then, for each background connection Â_{0} on 𝔤_{0}, we will define an upwards overlap map,

(6.29)  ρ_{P,[P′]}^{𝔱,𝔰,u} : O(𝔱, 𝔰, P, [P′], g_{P}) ⊂ Gl(𝔱, 𝔰, P, [P′], g_{P}) → Gl(𝔱, 𝔰, [P < P′], g_{P′}).

In words, the map ρ_{P,[P′]}^{𝔱,𝔰,u} will be defined by keeping the S^{4} connections, and parallel translating the frames (F_{P}^{T} , F_{P}^{𝔤}) from y_{P} ∈ X to each x_{Q} ∈ X, where (x_{Q})_{Q∈P′′} is the image of A under the composition e(X^{ℓ}, P) ∘ π_{ν(Σ)}.

We assume that the domain O(𝔱, 𝔰, P, [P′], g_{P}) of ρ_{P,[P′]}^{𝔱,𝔰,u} satisfies

(6.30)  O(𝔱, 𝔰, P, [P′], g_{P}) ⊂ π_{ν(Σ)}^{−1} (O(X^{ℓ}, P, [P′])) ,

where O(X^{ℓ}, P, [P′]) ⊂ ν(X^{ℓ}, P, [P′]) is the subspace mapped to the tubular neighborhood U(X^{ℓ}, P) in Lemma 6.4. Then, the map e(X^{ℓ}, P) ∘ π_{ν(Σ)} is defined on O(X^{ℓ}, P, [P′]) and takes values in U(X^{ℓ}, [P < P′]), the intersection of U(X^{ℓ}, P) with the diagonals Σ(X^{ℓ}, P′′) for P′′ ∈ [P < P′].

Assume x is a point in a geodesic ball around y ∈ X. Then for any connection A on a bundle over X and F a frame of the bundle lying over y, let T_{x,y}^{A}(F) denote parallel translation (parallel with respect to the connection A) of the frame F from x to y along the radial geodesic. We will use T_{x,y}^{LC} to denote parallel translation with respect to the Levi-Civita connection.

We now define the upwards overlap map. Assume A ∈ O(𝔱, 𝔰, P, [P′]) satisfies

π_{Σ}(A) = y = (y_{P})_{P ∈P} and e(X^{ℓ}, P) ∘ π_{ν(Σ)}(A) = (x_{Q})_{Q∈P′′}.

Then,

(6.31)  \begin{aligned}ρ_{P,[P′]}^{𝔱,𝔰,u}(A) & = ρ_{P,[P′]}^{𝔱,𝔰,u} ((F_{P}^{T} , F_{P}^{𝔤})_{P ∈P}, ((v_{Q})_{Q∈P′′_{P}} , ([A_{Q}, F_{Q}^{s}, x_{Q}])_{Q∈P′′_{P}} )_{P ∈P}) \\ & = ((T_{x_{Q},y_{P}}^{LC} (F_{P}^{T}), T Â_{0} ; x_{Q},y_{P} (F_{P}^{𝔤}))_{Q∈P′′}, ([A_{Q}, F_{Q}^{s}, x_{Q}])_{Q∈P′′_{P}} ) , \end{aligned}

where the indices P and Q appearing in the parallel translations T_{x_{Q},y_{P}}^{LC} and T_{x_{Q},y_{P}}^{A} satisfy Q ⊆ P. The Levi-Civita connection used to define T_{x_{Q},y_{P}}^{LC} is the metric g_{P,y}. The map

ρ_{P,[P′]}^{𝔱,𝔰,u} thus depends on the background connection Â_{0} used to define T Â_{0} ; x_{Q},y_{P}.

We define the subspace,

(6.32) \begin{matrix}T (𝔱, 𝔰, P, [P′]) \\ = \{((F_{P}^{T} , F_{P}^{𝔤})_{P ∈P}, ((v_{Q})_{Q∈P′′_{P}} , [Θ, F_{Q}^{s}, x_{Q}])_{Q∈P′′_{P}} )_{P ∈P}) ∈ Gl(𝔱, 𝔰, P, [P′])\}. \end{matrix}

That is, T (𝔱, 𝔰, P, [P′]) is the subspace of Gl(𝔱, 𝔰, P, [P′], g_{P}) where all the S^{4} connections are trivial. The proof of the following is trivial. Lemma 6.7. Let P < P′ be partitions of N_{ℓ}. Assume the open subspace O(𝔱, 𝔰, P, [P′], g_{P}) of Gl(𝔱, 𝔰, P, [P′], g_{P}) satisfies the condition (6.30) Then, for each connection Â_{0} on 𝔤_{0}, the map

ρ_{P,[P′]}^{𝔱,𝔰,u} : O(𝔱, 𝔰, P, [P′], g_{P}) → Gl(𝔱, 𝔰, [P < P′], g_{P′}),

defines an open embedding of O(𝔱, 𝔰, P, [P′], g_{P}) into Gl(𝔱, 𝔰, [P < P′], g_{P′}).

The upwards overlap map ρ_{P,[P′]}^{𝔱,𝔰,u} depends on the choice of a connection Â_{0} on 𝔤_{0}. For a

background pair (A_{0}, Φ_{0}) ∈ C̃_{𝔱_{0}}, we will write ρ_{P,[P′]}^{𝔱,𝔰,u}(A_{0}, Φ_{0}) for the upwards overlap map

defined by the connection Â_{0}. Then, we define

(6.33)  ρ_{[P<P′]}^{N,𝔱,𝔰,u} : Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P, [P′], g_{P}) → Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, [P < P′], g_{P′})

by, for (A_{0}, Φ_{0}) ∈ Ñ_{𝔱_{0},𝔰} and F ∈ O(𝔱, 𝔰, P, [P′], g_{P}),

ρ_{[P<P′]}^{N,𝔱,𝔰,u} ((A_{0}, Φ_{0}), F) = ((A_{0}, Φ_{0}), ρ_{P,[P′]}^{𝔱,𝔰,u}(Θ_{P}(A_{0}, Φ_{0}, π_{Σ}(F))(F))) .

In words, ρ_{[P<P′]}^{N,𝔱,𝔰,u} leaves the background pair alone and the restriction of ρ_{[P<P′]}^{N,𝔱,𝔰,u} to

G_{𝔰}(A_{0}, Φ_{0}) ×_{G_{𝔰}} O(𝔱, 𝔰, P, [P′], g_{P})|_{π_{Σ}^{−1}(y)}

is equal to the upwards overlap map ρ_{P,[P′]}^{𝔱,𝔰,u} given by the background connection Â′ where (A′, Φ′) = Θ_{P}(A_{0}, Φ_{0}, y).

6.4.3. The downwards overlap map. The downwards overlap map is a map,

ρ_{P,P′}^{𝔱,𝔰,d} : O_{d}(𝔱, 𝔰, P, [P′], g_{P}) ⊂ Gl(𝔱, 𝔰, P, [P′], g_{P}) → Gl(𝔱, 𝔰, P, g_{P}),

defined analogously to the map ρ_{P,P′}^{Θ,d}. We wish to define, for A ∈ Gl(𝔱, 𝔰, P, [P′], g_{P}),

(6.34)  \begin{aligned}ρ_{P,[P′]}^{𝔱,𝔰,d}(A) & = ρ_{P,[P′]}^{𝔱,𝔰,d} ((F_{P}^{T,𝔤})_{P ∈P}, ((v_{Q})_{Q∈P′′_{P}} , ([A_{Q}, F_{Q}^{s}, x_{Q}])_{Q∈P′′_{P}} )_{P ∈P}) \\ & = ((F_{P}^{T,𝔤})_{P ∈P}, (γ′_{ℝ^{4},P′′_{P}} ((v_{Q})_{Q∈P′′_{P}} , ([A_{Q}, F_{Q}^{s}, x_{Q}])_{Q∈P′′_{P}} ))) . \end{aligned}

In words, the map ρ_{P,[P′]}^{𝔱,𝔰,d} is defined by applying the splicing map γ′_{Θ,P′′_{P}} of (5.19) to the data in

(6.35)  Δ^{∘}(Z_{|P |}(δ_{P}), P′′_{P}) × ∏_{Q∈P′′_{P}} M̄_{spl,|Q|}^{s,♮}(δ_{Q}).

Thus, for the map ρ_{P,[P′]}^{𝔱,𝔰,d} to be defined, we must restrict ρ_{P,[P′]}^{𝔱,𝔰,d} to the points where the data in (6.35) lie in the domain of the splicing map γ′_{Θ,P′′_{P}}. More formally, we must assume that

domain of ρ_{P,[P′]}^{𝔱,𝔰,d} satisfies

(6.36)

O_{d}(𝔱, 𝔰, P, [P′], g_{P})
⊆ {((F_{P}^{T} , F_{P}^{𝔤})_{P ∈P}, ((v_{Q})_{Q∈P′′_{P}} , ([A_{Q}, F_{Q}^{s}, x_{Q}])_{Q∈P′′_{P}} )_{P ∈P}) ∈ Gl(𝔱, 𝔰, P, [P′], g_{P}) :
((v_{Q})_{Q∈P′′_{P}} , ([A_{Q}, F_{Q}^{s}, x_{Q}])_{Q∈P′′_{P}} ) ∈ O_{1}^{asd}(ℝ^{4}, P′′_{P}) for all P ∈ P},

where O_{1}^{asd}(Θ, P′′_{P} , δ_{P}) is the open set defined in Lemma 5.17. Lemma 6.8. Let O_{d}(𝔱, 𝔰, P, [P′], g_{P}) ⊆ Gl(𝔱, 𝔰, P, [P′], g_{P}) be any open neighborhood of Σ(𝔱, 𝔰, P, [P′], g_{P}) satisfying (6.36). Then the restriction of ρ_{P,[P′]}^{𝔱,𝔰,d} to O_{d}(𝔱, 𝔰, P, [P′], g_{P}) is an open embedding whose image is an open neighborhood of Σ(𝔱, 𝔰, P, g_{P}) in Gl(𝔱, 𝔰, P, g_{P}).

6.4.4. The splicing equality. We now show how the constructions of the overlap maps and crude splicing maps lead to a diagram similar to the diagram (5.52) used to construct the spliced end W_{κ}.

For P′′ ∈ [P < P′], let O(𝔱, 𝔰, P′′, g_{P′′}) ⊂ Gl(𝔱, 𝔰, P′′, g_{P′′}) be a 𝔖(P)-invariant collection of open subspaces on which the crude splicing maps γ′′_{X,P′′} are defined. Define (6.37)

O(𝔱, 𝔰, [P < P′], g_{P′}) = ( ∐_{P′′∈[P<P′]} O(𝔱, 𝔰, P′′, g_{P′′})) /𝔖(P) ⊂ Gl(𝔱, 𝔰, [P < P′], g_{P′}).

Then, by the 𝔖(P)-invariance of the crude splicing maps γ′′_{𝔱,𝔰,P′′} for P′′ ∈ [P < P′], the mapsγ′′_{𝔱,𝔰,P′′} define a crude splicing map

(6.38)  γ′′_{𝔱,𝔰,[P<P′]} : Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, [P < P′], g_{P′}) → C̄_{𝔱}.

Given subspaces, O(𝔱, 𝔰, P, g_{P}) and an 𝔖(P)-invariant collection of subspaces O(𝔱, 𝔰, P′′, g_{P′′}) for P′′ ∈ [P < P′], we define an open subspace,

(6.39)  O_{1}(𝔱, 𝔰, P, [P′], g_{P}) ⊂ Gl(𝔱, 𝔰, P, [P′], g_{P})

to be the points satisfying the conditions (6.30) and (6.36) and, in addition, satisfying

O_{1}(𝔱, 𝔰, P, [P′], g_{P}) ⊆ (ρ_{P,[P′]}^{𝔱,𝔰,u})^{−1} (O(𝔱, [P < P′])) ∩ (ρ_{P,[P′]}^{𝔱,𝔰,d})^{−1} (O(𝔱, 𝔰, P, g_{P})) .

Then we have, Proposition 6.9. Let P < P′ be partitions of N_{ℓ}. Assume the families of metrics g_{P′′} satisfy the conditions in §6.1.2. Let O(𝔱, 𝔰, P, g_{P} ⊂ Gl(𝔱, 𝔰, P, g_{P}) be an open subspace on which the crude splicing map is defined. Let {O(𝔱, 𝔰, P′′, g_{P′′})}_{P′′∈[P<P′]} be an 𝔖(P)- invariant collection of open subspaces on which the crude splicing maps, γ′′_{𝔱,𝔰,P′′} are defined and let O(𝔱, 𝔰, [P < P′], g_{P′}) be the resulting space defined by (6.37). Write id_{N} for the identity map on Ñ = Ñ_{𝔱_{0},𝔰}. Then, if O_{1}(𝔱, 𝔰, P, [P′], g_{P}) is the subspace defined in (6.39), the maps in the diagram

(6.40)  \begin{matrix}Ñ(δ) ×_{G_{𝔰}} O_{1}(𝔱, 𝔰, P, [P′], g_{P}) & ⟶^{ρ_{P,[P′]}^{N,𝔱,𝔰,u}} & Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, [P < P′], g_{P′}) \\ ^{id_{N} ×ρ_{P,[P′]}^{𝔱,𝔰,d}}↓ & & ^{γ}′′_{𝔱,𝔰,[P<P′]} ↓ \\ Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P}) & ⟶^{γ′′_{𝔱,𝔰,P}} & C̄_{𝔱} \end{matrix}

are S^{1}-equivariant and the diagram commutes.

Proof. The conditions in the definition of the subspace O_{1}(𝔱, 𝔰, P, [P′], g_{P}) imply that the compositions

γ′′_{𝔱,𝔰,[P<P′]} ∘ ρ_{P,[P′]}^{N,𝔱,𝔰,u} and γ′′_{𝔱,𝔰,P} ∘ (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d})

are defined on Ñ(δ) ×_{G_{𝔰}} O_{1}(𝔱, 𝔰, P, [P′], g_{P}).

The S^{1}-equivariance follows from the S^{1}-equivariance of the crude splicing maps and the observation that the overlap maps in (6.40) are S^{1}-equivariant because the S^{1} action is simply scalar multiplication on the section in Ñ_{𝔱_{0},𝔰} and the overlap maps do not change the background section.

To see that the diagram (6.40) commutes, begin by restricting the diagram to

G_{𝔰}(A_{0}, Φ_{0}) ×_{G_{𝔰}} O(𝔱, 𝔰, P, [P′], g_{P})|_{π_{Σ}^{−1}(y)} ⊂ Ñ ×_{G_{𝔰}} O(𝔱, 𝔰, P, [P′], g_{P})

for (A_{0}, Φ_{0}) ∈ Ñ and y ∈ Δ^{∘}(X^{ℓ}, P). The definition of the overlap maps imply that the crude splicing maps are defined by splicing in S^{4} connections at y for γ′′_{𝔱,𝔰,P} and at some y′ for γ′′_{X,[P<P′} where π(X^{ℓ}, P)(y′) = y. Then, the consistency of the families of metrics g_{P} and g_{P′} imply g_{P,y} = g_{P′,y′}. Similarly, the consistency condition of the flattening maps in (6.17) imply that (A′_{0}, Φ′_{0}) = Θ_{P}(A_{0}, Φ_{0}, y) = Θ_{P′}(A_{0}, Φ_{0}, y′). This proves that the sections obtained by the two compositions in the diagram (6.40) are equal to Φ′_{0}. The connections obtained by the two compositions in the diagram (6.40) are defined by splicing connections on S^{4} to the background connection A′_{0}.

The property (6.30) of O(𝔱, 𝔰, P, [P′], g_{P}) implies that y′ lies in the suitably small tubular neighborhood U(X^{ℓ}, P) defined in Lemma 6.4. Thus, from (6.19) the inclusions

B(y′_{P′}, 4s̃_{P′}(y′)^{1/2}) ⋐ B(y_{P} , 2s̃_{P} (y)^{1/2})

hold for the balls where the splicing takes place. Then because the connection Â′_{0} and metric g_{P,y} are flat on each ball B(y_{P} , 2s̃_{P} (y)^{1/2}), the proof of the proposition reduces to Lemma 5.11. □

6.5. Constructing the space of global splicing data. The construction of the space of global splicing data follows from Proposition 6.9 and the arguments on shrinking the neighborhoods O(𝔱, 𝔰, P, g_{P}) given in the construction of the spliced end W_{κ} in §5.4.

Theorem 6.10. For every partition P of N_{ℓ}, there is a neighborhood O(𝔱, 𝔰, P, g_{P}) of Σ(𝔱, 𝔰, P, g_{P}) in Gl(𝔱, 𝔰, P, g_{P}) such that the intersection of the images

γ′′_{𝔱,𝔰,P}(O(𝔱, 𝔰, P, g_{P})) ∩ γ′′_{P′}(O(𝔱, 𝔰, P′, g_{P′}))

is empty if P ≮ P′ and P′ ≮ P and is contained in

γ′′_{𝔱,𝔰,P} (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d} (Ñ ×_{G_{𝔰}} O_{1}(𝔱, 𝔰, P, [P′], g_{P})))
= γ′′_{[P<P′]} (ρ_{P,[P′]}^{N,d} (Ñ ×_{G_{𝔰}} O_{1}(𝔱, 𝔰, P, [P′], g_{P})))

if P < P′, where O_{1}(𝔱, 𝔰, P, [P′], g_{P}) is the subspace appearing in Proposition 6.9.

Proof. The proof is identical to the arguments appearing in Lemma 5.17 for the construction of W_{κ}. □

We then define the space of global splicing data to be

(6.41)  M̄_{𝔱,𝔰}^{vir} = ⊔_{P} (Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P})) / ∼

where O(𝔱, 𝔰, P, g_{P}) is the space appearing in Theorem 6.10 and points in

Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P}) and Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P′, g_{P′})

are identified by the relation ∼ if their images under the crude splicing maps are equal. Thus, the space M̄_{𝔱,𝔰}^{vir} is the homotopy pushout of the diagram (6.40) in the sense of the following lemma.

Lemma 6.11. If P < P′ are partitions of N_{ℓ}, (A_{0}, Φ_{0}, F) ∈ Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P}) and (A′_{0}, Φ′_{0}, F′) ∈ Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P′, g_{P′}), then

(A_{0}, Φ_{0}, F) ∼ (A′_{0}, Φ′_{0}, F′)

if and only if (A_{0}, Φ_{0}) = (A′_{0}, Φ′_{0}) and there is (A_{0}, Φ_{0}, A) ∈ Ñ ×_{G_{𝔰}} O_{1}(𝔱, 𝔰, P, [P′], g_{P}) satisfying

(6.42)

(A_{0}, Φ_{0}, F) = (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d})(A_{0}, Φ_{0}, F_{0})
(A′_{0}, Φ′_{0}, F′) = ρ_{P,[P′]}^{N,d}(A_{0}, Φ_{0}, F_{0}).

Because the overlaps of the images of γ′′_{𝔱,𝔰,P} are controlled by the diagram (6.40) and because the maps id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d} and ρ_{P,[P′]}^{N,𝔱,𝔰,u} are smoothly-stratified embeddings, we have the following.

Corollary 6.12. The space M̄_{𝔱,𝔰}^{vir} is a smoothly-stratified space admitting an S^{1}-equivariant fibration,

(6.43)  π_{N} : M̄_{𝔱,𝔰}^{vir} → N_{𝔱_{0},𝔰}(δ),

and an S^{1}-equivariant embedding

(6.44)  γ′′_{M} : M̄_{𝔱,𝔰}^{vir} → C̄_{𝔱},

such that the restriction of γ′′_{M} to Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P}) is equal to γ′′_{𝔱,𝔰,P}.

6.6. Thom-Mather structures on the space of global splicing data. On each open subspace

U(𝔱, 𝔰, P) = Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P}) ⊂ M̄_{𝔱,𝔰}^{vir},

there is a projection map

(6.45)  π(𝔱, 𝔰, P) : U(𝔱, 𝔰, P) → N(δ) × Σ(𝔱, 𝔰, P, g_{P})

defined by id_{N} × π_{Σ}, where π_{Σ} is the restriction of the projection defined in (6.10) to O(𝔱, 𝔰, P, g_{P}) and id_{N} is the identity on Ñ(δ).

To prove the relations (4.1) holds for the projection maps π(𝔱, 𝔰, P) we make the following observations on the overlap maps. First, note that the projection maps,

π(Θ, P′_{P}) : U(Θ, P′_{P}) ⊂ M̄_{spl,|P |}^{s,♮}(δ_{P}) → {[Θ]} × Σ(Z_{P} (δ_{P}), P′_{P})

defined in (5.66) are SO(3) × SO(4) equivariant. Together with the identify map id_{N} on Ñ(δ), the maps π(Θ, P′′_{P}) (for P′′ ∈ [P < P′]) define a map, id_{N} × p_{P,[P′]},

(6.46)

Fr(𝔱, 𝔰, P, g_{P}) ×_{G̃(P)} ∏_{P ∈P} ∐_{P′′∈[P<P′]} U(Θ, P′′_{P})
id_{N} ×p_{P,[P′]}↓
Fr(𝔱, 𝔰, P, g_{P}) ×_{G̃(P)} ∏_{P ∈P} ∐_{P′′∈[P<P′]}{[Θ]} × Σ(Z_{P} (δ_{P}), P′′_{P})

Note that because

∐_{P′′∈[P<P′]} U(Θ, P′′_{P}) ⊂ M̄_{spl,|P |}^{s,♮}(δ_{P}), and

∐_{P′′∈[P<P′]}{[Θ]} × Σ(Z_{P} (δ_{P}), P′′_{P}) ⊂ M̄_{spl,|P |}^{s,♮}(δ_{P})

the map id_{N} × p_{P,P′} is defined on a subspace of Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P}). Observe that the relation

(6.47)  (id_{N} × π_{Σ}) ∘ (id_{N} × p_{P,P′}) = (id_{N} × π_{Σ})

holds where π_{Σ} is the projection given in (6.10) because p_{P,P′} respects the fibers of Gl(𝔱, 𝔰, P, g_{P}).

Then for P < P′, the following lemma describes the projection map π(𝔱, 𝔰, P′) on the image of the overlap map ρ_{P,[P′]}^{𝔱,𝔰,d}.

Lemma 6.13. Let P < P′ be partitions of N_{ℓ}. For π_{Σ,P,[P′]} the projection defined in (6.27), let

id_{N} × π_{Σ,P,[P′]} : Ñ(δ) ×_{G_{𝔰}} Gl(𝔱, 𝔰, P, [P′], g_{P}) → N(δ) × Σ(𝔱, 𝔰, P, [P′], g_{P})

be the product of π_{Σ,P,[P′]} with the identity map on Ñ(δ). Then, the relations

(6.48)

π(𝔱, 𝔰, P′) ∘ ρ_{P,[P′]}^{N,𝔱,𝔰,u} = ρ_{P,[P′]}^{N,𝔱,𝔰,u} ∘ (id_{N} × π_{Σ,P,[P′]})
(id_{N} × p_{P,[P′]}) ∘ (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d}) = (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d}) ∘ (id_{N} × π_{Σ,P,[P′]})

hold on Ñ ×_{G_{𝔰}} O_{1}(𝔱, 𝔰, P, [P′], g_{P}).

We now prove the first Thom-Mather relation between the projections. Lemma 6.14. Let P < P′ be partitions of N_{ℓ}. Then, restricted to the intersection

U(𝔱, 𝔰, P) ∩ U(𝔱, 𝔰, P′),

the equality

(6.49)  π(𝔱, 𝔰, P) ∘ π(𝔱, 𝔰, P′) = π(𝔱, 𝔰, P)

holds.

Proof. By Theorem 6.10, the intersection U(𝔱, 𝔰, P) ∩ U(𝔱, 𝔰, P′) is contained in the image of the overlap maps in the diagram (6.40). Therefore every point A′ in the intersection is given by

A′ = (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d})(A) = ρ_{P,[P′]}^{N,𝔱,𝔰,u}(A).

for some A ∈ Ñ ×_{G_{𝔰}} O_{1}(𝔱, 𝔰, P, [P′], g_{P}). Then

π(𝔱, 𝔰, P′)(A′) = π(𝔱, 𝔰, P′) ∘ ρ_{P,[P′]}^{N,𝔱,𝔰,u}(A)
= ρ_{P,[P′]}^{N,𝔱,𝔰,u} ∘ (id_{N} × π_{Σ,P,[P′]})(A) By equation (6.48)
= (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d}) ∘ (id_{N} × π_{Σ,P,[P′]})(A) By the definition of ∼
= (id_{N} × p_{P,[P′]}) ∘ (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d})(A) By equation (6.48)

The preceding equality implies

(π(𝔱, 𝔰, P) ∘ π(𝔱, 𝔰, P′))(A′)

= (π(𝔱, 𝔰, P) ∘ (id_{N} × p_{P,[P′]}) ∘ (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d}))(A)

= ((id_{N} × π_{Σ}) ∘ (id_{N} × p_{P,[P′]}) ∘ (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d}))(A) By the definition following (6.45)

= (id_{N} × π_{Σ}) ∘ (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d}))(A) By equation (6.47)

= (π(𝔱, 𝔰, P) ∘ (id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d}))(A) By the definition following (6.45)

= (π(𝔱, 𝔰, P)(A′),

thus proving (6.49). □

We now define a tubular distance function,

t(𝔱, 𝔰, P) : O(𝔱, 𝔰, P) → [0, 1)

which is invariant under the actions of G_{𝔰} and S^{1} on Gl(𝔱, 𝔰, P, g_{P}) and which satisfies:

t(𝔱, 𝔰, P)^{−1}(0) = Σ(𝔱, 𝔰, P, g_{P}).

The natural choice is

(6.50)  t(𝔱, 𝔰, P) = ∑_{P ∈P} |P |λ̃_{P}^{2} ,

where λ̃_{P} is the modified scale parameter defined in (5.69). We shall also use a rescaling of this map.

We note that all maps defined on subspaces of Sym^{ℓ}(X) define maps on the subspace N(δ) × Sym^{ℓ}(X) by pullback. Lemma 6.15. There are tubular distance functions,

(6.51)  t(𝔱, 𝔰, P) : U(𝔱, 𝔰, P) → [0, 1), and t_{r}(𝔱, 𝔰, P) : U(𝔱, 𝔰, P) → [0, 1)

satisfying (1) t(𝔱, 𝔰, P)^{−1}(0) = t_{r}(𝔱, 𝔰, P)^{−1}(0) = N(δ) × Σ(X^{ℓ}, P), (2) The restriction of t(𝔱, 𝔰, P) to the subspace

(N(δ) × U′′(X^{ℓ}, P, g_{P})) ∩ U(𝔱, 𝔰, P)

is equal to the pullback of the map t(X^{ℓ}, P) defined in (4.37) under the obvious projection,

(3) On the intersection U(𝔱, 𝔰, P) ∩ U(𝔱, 𝔰, P′) the relations,

(6.52)  t(𝔱, 𝔰, P) ∘ π(𝔱, 𝔰, P′) = t(𝔱, 𝔰, P) and t_{r}(𝔱, 𝔰, P) ∘ π(𝔱, 𝔰, P′) = t_{r}(𝔱, 𝔰, P)

hold.

(4) There are open neighborhoods

U_{1}(𝔱, 𝔰, P) ⋐ U_{2}(𝔱, 𝔰, P) ⋐ U(𝔱, 𝔰, P)

of N(δ) × Σ(X^{ℓ}, P), such that

(6.53)  \begin{aligned}U_{1}(𝔱, 𝔰, P) & = t_{r}(𝔱, 𝔰, P)^{−1}([0, \frac{1}{2})), \\ U(𝔱, 𝔰, P) − U_{2}(𝔱, 𝔰, P) & = t_{r}(𝔱, 𝔰, P)^{−1}(\{1\}) \end{aligned}

Proof. The first and second items follow immediately from the definition. To prove the equality (6.52) for the function t(𝔱, 𝔰, P) defined in (6.50), we begin by recalling that the modified scale function λ̃_{P} satisfied

λ̃_{P} ∘ π(Θ, P′′_{P}) = λ̃_{P} .

Then the definition of the projection id_{N} × p_{P,[P′]} in (6.46) implies

t(𝔱, 𝔰, P) ∘ (id_{N} × p_{P,[P′]}) = t(𝔱, 𝔰, P).

The equality (6.52) then follows by the argument in Lemma 6.14.

The function t_{r}(𝔱, 𝔰, P) is defined by a rescaling of t(𝔱, 𝔰, P) (i.e. multiplying by a function pulled back from N(δ) × Σ(X^{ℓ}, P)) which does not affect the property (6.52). The third item then follows by the argument given for item (6) in Lemma 5.20. □

6.7. The global splicing map. The global crude splicing map γ′′_{M} of (6.44) is unsatisfactory because its image does not include M_{𝔰} × Sym^{ℓ}(X) (see Lemma 6.5 and Remark 6.6). To remedy this, we define an isotopy of the embedding γ′′_{M}.

First, observe that if β : [0, 1] → [0, 1] is a smooth function with β([0, \frac{1}{4}]) = 1 and β([\frac{1}{2}, 1]) = 0, then β ∘ t_{r}(𝔱, 𝔰, P) is supported in U_{1}(𝔱, 𝔰, P) and vanishes on U(𝔱, 𝔰, P) − U_{2}(𝔱, 𝔰, P) where the spaces U_{i}(𝔱, 𝔰, P) are given in appearing in Lemma 6.15. Hence, the function

β_{P} = β ∘ t_{r}(𝔱, 𝔰, P)

can be considered a global function on M̄_{𝔱,𝔰}^{vir} satisfying:

(6.54)  M̄_{𝔱,𝔰}^{vir} − U_{2}(𝔱, 𝔰, P) ⊂ β_{P}^{−1}(0) and U_{1}(𝔱, 𝔰, P) ⊂ β_{P}^{−1}(1).

Pick one representative P of each conjugacy class [P] and enumerate these representatives in such a way that

Σ(X^{ℓ}, P_{i}) ⊂ cl_{Sym^{ℓ}(X)}Σ(X^{ℓ}, P_{j})

implies i < j. Assume P_{1} = {N_{ℓ}} is the crudest partition. We define a finite sequence of embeddings,

γ′′_{M,i} : M̄_{𝔱,𝔰}^{vir} → C̄_{𝔱},

by, if γ′_{𝔱,𝔰,P} is the standard splicing map defined on U(𝔱, 𝔰, P),

(6.55)  \begin{aligned}γ′′_{M,0} & = γ′′_{M}, \\ γ′′_{M,i} & = β_{P}(·)γ′_{𝔱,𝔰,P}(·) + (1 − β_{P})(·)γ′′_{M,i−1}(·). \end{aligned}

Observe that although γ′_{𝔱,𝔰,P} is only defined on the subspace U(𝔱, 𝔰, P) ⊂ M̄_{𝔱,𝔰}^{vir}, the convex combination of the embeddings γ′_{𝔱,𝔰,P} and γ′′_{M,i−1} is defined on all of M̄_{𝔱,𝔰}^{ir} because β_{P} vanishes on the complement of U(𝔱, 𝔰, P). By (6.54), γ′′_{M,i} is equal to the standard splicing map γ′_{𝔱,𝔰,P} on U_{1}(𝔱, 𝔰, P) and equal to γ′′_{M,i−1} on M̄_{𝔱,𝔰}^{vir} − U_{2}(𝔱, 𝔰, P). Assume that P_{r} = N_{ℓ} is the finest partition. We then define

(6.56)  γ′_{M} = γ′′_{M,r},

It is straightforward to verify that

(1) The map γ′_{M} is a continuous S^{1}-equivariant embedding and is smooth on each stratum,

(2) The restriction of γ′_{M} to N(δ) × Sym^{ℓ}(X) is equal to the product of the embedding N(δ) → C_{𝔱_{0}} with the identity on Sym^{ℓ}(X).

6.8. Projection to Sym^{ℓ}(X). We now define a projection map,

(6.57)  π_{X} : M̄_{𝔱,𝔰}^{vir} → Sym^{ℓ}(X),

which will be used in describing cohomology classes on M̄_{𝔱,𝔰}^{vir}.

We will denote the projection π_{Σ} of (6.10) by

π_{Σ}(𝔱, 𝔰, P) : U(𝔱, 𝔰, P) → Σ(X^{ℓ}, P),

to avoid confusion between the projection maps of the different strata. These local projections maps do not agree on the intersections U(𝔱, 𝔰, P) ∩ U(𝔱, 𝔰, P′) but can be related by the following. Lemma 6.16. If P < P′ are partitions of N_{ℓ}, then

(6.58)  π(X^{ℓ}, P) ∘ π_{Σ}(𝔱, 𝔰, P′) = π_{Σ}(𝔱, 𝔰, P)

on U(𝔱, 𝔰, P) ∩ U(𝔱, 𝔰, P′).

Proof. The lemma follows immediately from Lemma 6.11 and the definitions of the projection maps. □

For P a partition of N_{ℓ}, let

r_{P} : U(X^{ℓ}, P) × [0, 1] → U(X^{ℓ}, P),

be the deformation retraction of U(X^{ℓ}, P) to Σ(X^{ℓ}, P) defined by identifying U(X^{ℓ}, P) with the normal bundle ν(X^{ℓ}, P) by the exponential map e(X^{ℓ}, P, g_{P}) and fiber multiplication by the parameter t ∈ [0, 1]. Thus, r_{P}(·, 1) is the identity map and r_{P}(·, 0) = π(X^{ℓ}, P, g_{P}). Because r_{P′}(·, 1) is the identity, we can define

r_{P} (π_{Σ}(𝔱, 𝔰, P′)(·), 1 − β_{P}(·)) : U(𝔱, 𝔰, P′) → Σ(X^{ℓ}, P′) ∪ Σ(X^{ℓ}, P).

Observe that on U_{1}(𝔱, 𝔰, P) ∩ U(𝔱, 𝔰, P′) the preceding map is equal to π(X^{ℓ}, P) ∘ π_{Σ}(𝔱, 𝔰, P′) which by Lemma 6.16 is equal to π_{Σ}(𝔱, 𝔰, P). We now prove the existence of the global projection π_{X} on an open subspace of M̄_{𝔱,𝔰}^{vir}.

Lemma 6.17. There is an open neighborhood U_{1} ⊂ M̄_{𝔱,𝔰}^{vir} of N(δ) × Sym^{ℓ}(X) and a projection map,

π_{X} : U_{1} → Sym^{ℓ}(X),

such that the restriction of π_{X} to U_{1} ∩ U(𝔱, 𝔰, P) is homotopic to π_{Σ}(𝔱, 𝔰, P).

Proof. We construct the map

π̃_{Σ}(𝔱, 𝔰, P′) : U(𝔱, 𝔰, P′) → clΣ(X^{ℓ}, P′)

inductively so that (1) The map π̃_{Σ}(𝔱, 𝔰, P′) is homotopic to π_{Σ}(𝔱, 𝔰, P),

(2) For all P < P′, on U_{1}(𝔱, 𝔰, P) ∩ U_{1}(𝔱, 𝔰, P), we have π̃_{Σ}(𝔱, 𝔰, P′) = π̃_{Σ}(𝔱, 𝔰, P). The open neighborhood U_{1} refered to in the statement of the lemma will be the union of U_{1}(𝔱, 𝔰, P′) and the map π_{X} will be defined by π̃_{Σ}(𝔱, 𝔰, P) on each subspace U_{1}(𝔱, 𝔰, P).

Let P_{0} = {N_{ℓ}} be the crudest partition of N_{ℓ}. Let π̃_{Σ}(𝔱, 𝔰, P_{0}) = π_{Σ}(𝔱, 𝔰, P_{0}).

Assume π̃_{Σ}(𝔱, 𝔰, P) has been constructed for all P < P′. The projection π̃_{Σ}(𝔱, 𝔰, P) is constructed by induction on P′ with P < P′ using the deformation r_{P′} and the parameter given by the function t_{r}(𝔱, 𝔰, P′) to homotope between π_{Σ}(𝔱, 𝔰, P) and a new map π̃_{Σ}(𝔱, 𝔰, P′) equal to π̃(𝔱, 𝔰, P′). The inductive assumption ensures there is no conflict for different P′. Let π̃_{Σ}(𝔱, 𝔰, P) be the end of this string of homotopies. □

7. The spliced-end obstruction bundle

The gluing map, described in Theorem 8.2, does not map all points in M̄_{𝔱,𝔰}^{vir} to the moduli space M̄_{𝔱}. Instead, there is a section (the obstruction section) of a pseudo-vector bundle (the obstruction bundle) over M̄_{𝔱,𝔰}^{vir} such that the intersection of M_{𝔱} with the image of the gluing map is given by the image of the zero-locus of the obstruction section.

The obstruction bundle has two components: the background obstruction and the instanton obstruction. In this section, we define a pseudo-bundle, the spliced-end obstruction bundle, Ῡ_{spl,κ}^{i} → M̄_{spl,κ}^{s,♮}(δ) which will be used to define the instanton obstruction over M̄_{𝔱,𝔰}^{vir}.

The restriction of Ῡ_{spl,κ}^{i} to the top stratum of M̄_{spl,κ}^{s,♮}, Υ_{spl,κ}^{i} → M_{spl,κ}^{s,♮} is a finitedimensional subbundle of the infinite-dimensional vector bundle,

(7.1)  𝔙_{κ} = A_{κ}^{♮}(2δ) × Fr(E_{κ})|_{s} ×_{G_{κ}} L_{k−1}^{2}(V_{κ}^{−}) → B_{κ}^{s}(2δ),

restricted to M_{spl,κ}^{s,♮}(δ), where V_{κ}^{±} = S^{±} ⊗ E_{κ} for the standard spin structure (𝔖^{±}, ρ) on S^{4}. The pseudo-bundle Ῡ_{spl,κ}^{i} will be a sub-bundle of the space

(7.2)  𝔙̄_{κ} = ⊔_{ℓ=0}^{κ}𝔙_{κ−ℓ} × Sym^{ℓ}(X),

but this extension will only be a pseudo-bundle rather than as a vector bundle as the rank of the extension will decrease on the lower strata.

The bundle Υ_{spl,κ}^{i} is an approximation to the index bundle Index(D_{κ}) associated to the Dirac operator D_{A} : Γ(V_{κ}^{−}) → Γ(V_{κ}^{+}) (as defined in §7.1) but an approximation that works more conveniently with the splicing construction of the space of global splicing data. To use the Ῡ_{spl,κ}^{i} to define an obstruction bundle over the space of splicing data, the Ῡ_{spl,κ}^{i} must be equivariant with respect to a Spin^{u}(4) action described in §7.2, just as the spliced ends moduli space is closed under the SO(3) × SO(4) action. The construction of the Ῡ_{spl,κ}^{i} is analogous to the that of the spliced-ends moduli space. Using induction, we require the restriction of Ῡ_{spl,κ}^{i} to the image of the splicing map γ′_{Θ,P} to be given by the image of a splicing map for sections of (7.2). An inductive argument, analogous to that used in constructing the subspace W_{κ} in §5.4. Then, following the argument in §5.6, the resulting bundle over W_{κ} is isotoped to one which is equal to Index(D_{κ}^{∗}) away from the trivial strata. These last arguments, because they are so similar to those in §5 are carried out in a compressed fashion in §7.3; the properties of Ῡ_{spl,κ}^{i} are enumerated in Proposition 7.1.

7.1. The equivariant Dirac index bundle. We begin by defining the bundle Index(D_{κ}^{∗}). Let 𝔰_{S^{4}} = (ρ, S^{+}, S^{−}) be the standard spin structure on S^{4} with respect to the round metric. Let E_{κ} → S^{4} be the rank-two, complex Hermitian bundle with c_{2}(E_{κ}) = κ. For [A] ∈ M_{κ}^{s,♮}(δ) we define the Dirac operator

(7.3)  D_{A} : L_{k}^{2}(S^{+} ⊗ E_{κ}) → L_{k−1}^{2}(S^{−} ⊗ E_{κ}),

to be the Dirac operator defined by the Levi-Civita connection on 𝔰_{S^{4}} and A on E_{κ}. The spliced-end obstruction bundle is an approximation to the bundle

Index(D_{κ}^{∗}) → M_{κ}^{s,♮}(δ),

defined by the cokernels of the Dirac operators:

(7.4) Index(D_{κ}^{∗}) = {(A, F^{s}, Ψ) ∈ M̃_{κ}^{s,♮} × Fr(E_{κ})|_{s} × L_{k−1}^{2}(S^{−} ⊗ E_{κ}) : D_{A}^{∗}Ψ = 0}/G_{κ}.

Because of the positive scalar curvature on S^{4}, the Weitzenbock estimates in [14] imply that the cokernel of D_{A}^{∗} vanishes for [A] ∈ M_{κ}^{s,♮}(δ) and thus Index(D_{κ}^{∗}) actually forms a vector bundle. An index theorem computation shows that Index(D_{κ}^{∗}) is a complex, rank κ vector bundle. We will also use the notation Index(D_{κ}^{∗}) to denote the bundle over M̄_{κ}^{s,♮}(δ) given by,

{[A, F^{s}, x, Ψ] : [A, F^{s}, x] ∈ M̄_{κ}^{s,♮}(δ) and D_{A}Ψ = 0}.

Because the index of the Dirac operator depends on κ, this extension of Index(D_{κ}^{∗}) is only a pseudo-vector bundle, that is, the rank of the fiber depends on the stratum.

7.2. The action of Spin^{u}(4). The group

Spin^{u}(4) = (U(2) × Spin^{c}(4)) /S^{1}

is defined in [11, §3.2] as are a pair of surjective homomorphisms

Ad_{SO(3)}^{u} : Spin^{u}(4) → SO(3), Ad_{SO(4)}^{u} : Spin^{u}(4) → SO(4)

(see [11, Equation (3.13)]). The bundle 𝔙_{κ} → B_{κ}^{s} is an equivariant Spin^{u}(4) bundle with respect to the action of Spin^{u}(4) on B_{κ}^{s} defined by the homomorphism

Spin^{u}(4) → SO(3) × SO(4), A ↦ (Ad(A), Ad(A)).

The bundle Index(D_{κ}^{∗}) is invariant under this action and thus is also an Spin^{u}(4) equivariant bundle.

7.3. The splicing map. The spliced-ends obstruction bundle will be identical to the index bundle Index(D_{κ}^{∗}) on the complement of the spliced ends of M̄_{spl,κ}^{s,♮}(δ). On the spliced end, W_{κ}, the spliced-ends obstruction bundle will be given by the image of a splicing map which we now introduce.

Recall that the end of M̄_{spl,κ}^{s,♮}(δ) near the trivial stratum,

{[Θ]} × Σ(Z_{κ}(δ), P)

where P is a partition of N_{κ} with |P| > 1, was given by the image of a splicing map

γ′_{Θ,P} : O^{asd}(Θ, P) ⊂ Δ^{∘}(Z_{κ}(δ), P) ×_{𝔖(P)} ∏_{P ∈P} M̄_{spl,|P |}^{s,♮}(δ_{P}) → M̄_{spl,κ}^{s,♮}(δ).

We will define the spliced-ends obstruction bundle inductively.

An element of the space 𝔙̄_{κ} from (7.2) can be written as [A, F^{s}, x, Ψ] where [A, F^{s}, x] ∈ B̄_{κ}^{s}(2δ), x ∈ Sym^{ℓ}(ℝ^{4}), and Ψ ∈ L_{k−1}^{2}(S^{−} ⊗ E_{κ−ℓ}). Then, for each partition P of N_{κ} we define an embedding, covering the splicing map γ_{Θ,P} by, (7.5)

\begin{matrix}φ′_{Θ,P} : Δ^{∘}(Z_{κ}(δ), P) ×_{𝔖(P)} ∏_{P ∈P} 𝔙̄_{|P |} → 𝔙̄_{κ}, \\ φ′_{Θ,P} ([x_{P} , [A_{P} , F_{P}^{s} , v_{P} , Ψ_{P}]]_{P ∈P}) = (γ′_{Θ,P}([x_{P} , [A_{P} , F_{P}^{s} , v_{P}]_{P ∈P}), ∑_{P ∈P} χ_{x_{P} ,λ_{P}} c_{x_{P} ,1}^{∗}Ψ_{P} ) . \end{matrix}

Here λ_{P} = λ([A_{P} , F_{P}^{s} , x_{P}]) and χ_{x,λ} is a cut-off function supported on B(x, \frac{1}{4}λ^{1/3}) and equal to one on B(x, \frac{1}{8}λ^{1/2}). Observe that the map (7.5) is equivariant with respect to the Spin^{u}(4) action if Spin^{u}(4) acts diagonally on the domain, by the projection Ad_{SO(4)}^{u} to SO(4) on the factor Δ^{∘}(Z_{κ}(δ), P). With this vocabulary established, we can now state the proposition detailing the properties of the spliced-end obstruction bundle. Proposition 7.1. There is a pseudo-bundle

(7.6)  Ῡ_{spl,κ}^{i} → M̄_{spl,κ}^{s,♮}(δ)

which is a Spin^{u}(4)-invariant subbundle of the space 𝔙̄_{κ} → B̄_{κ}^{s,♮}(2δ) and satisfies

(1) The restriction of Ῡ_{spl,κ}^{i} to the top stratum, M_{spl,κ}^{s,♮}(δ), is a complex, rank κ vector bundle,

(2) For any partition P of κ with |P| > 1, the restriction of Ῡ_{spl,κ}^{i} to the neighborhood of {[Θ]} × Σ(Z_{κ}(δ), P) given in (5.3) is given by the image of

(7.7)  Δ^{∘}(Z_{κ}(δ), P) ×_{𝔖(P)} ∏_{P ∈P} Ῡ_{spl,|P |}^{i},

under the splicing map φ′_{Θ,P},

(3) The restriction of Ῡ_{spl,κ}^{i} to the complement of W_{κ} is given by the restriction of the bundle Index(D_{κ}^{∗}) defined in (7.4) to the complement of W_{κ},

(4) For any [A′, F^{s}, x] ∈ M̄_{spl,κ}^{s,♮}(δ), L^{2}-orthogonal projection defines an isomorphism,

Υ_{spl,κ}^{i}|_{[A,F^{s},x]} ≅ Coker D_{A_{t}}

for all t ∈ [0, 1] where A_{t} = A′ + ta and A = A′ + a is the value of the gluing map at A′.

The construction of the bundle Ῡ_{spl,κ}^{i} uses induction on κ and is parallel to the construction of the spliced-ends moduli space. Thus, the first step (done in §7.3.1) is show that the images of the splicing maps φ′_{Θ,P} form a vector bundle over W_{κ}. We then use the final property in the proposition property that L^{2}-orthogonal projection gives an isomorphism to define the isotopy between the image of the splicing maps over W_{κ} and the bundle Index(D_{κ}^{∗}).

7.3.1. The overlap bundles. If the restriction of the spliced end index bundle to the image of the splicing map γ′_{Θ,P} is to be given by the image of the splicing map φ′_{Θ,P} restricted to the subspace (7.7), then we need to describe the overlaps of the images of the splicing maps φ′_{Θ,P} and φ′_{Θ,P′} where P < P′. This is done by constructing a space of overlap data analogous to that appearing in Proposition 5.12.

Begin by initially considering the vector pseudo-bundle Υ̃(Θ, P, P′) defined by restricting the vector bundle

(7.8)  Δ^{∘}(Z_{κ}(δ), P) × ∏_{P ∈P} (Δ^{∘}(Z_{|P |}(δ_{P}), P′_{P}) × ∏_{Q∈P′_{P}} Ῡ_{spl,|Q|}^{i}) ; ↓ ; Δ^{∘}(Z_{κ}(δ), P) × ∏_{P ∈P} (Δ^{∘}(Z_{|P |}(δ_{P}), P′_{P}) × ∏_{Q∈P′_{P}} M̄_{spl,|Q|}^{s,♮}(δ_{Q}))

to the intersection of its base with the space Õ(Θ, P, P′, δ) defined in Lemma 5.10. By induction on κ, the splicing maps φ_{Θ,P′_{P}} define maps

φ_{Θ,P′_{P}} : Δ^{∘}(Z_{|P |}(δ_{P}), P′_{P}) × ∏_{Q∈P′_{P}} Υ_{spl,|Q|}^{i} → Υ_{spl,|P |}^{i},

covering the splicing map γ′_{Θ,P′_{P}}. Hence, there is a map

ρ_{P,P′}^{Ψ,d} : Υ̃(Θ, P, P′) → Υ(Θ, P) = Δ^{∘}(Z_{κ}(δ), P) × ∏_{P ∈P} Υ_{spl,|P |}^{i}

defined by

ρ_{P,P′}^{Ψ,d} = id_{Δ} × ∏_{P ∈P} φ′_{Θ,P′_{P}} ,

where id_{Δ} is the identity map on Δ^{∘}(Z_{κ}(δ), P). By the definition of the overlap map ρ_{P,P′}^{Θ,d} in (5.34), the map ρ_{P,P′}^{Ψ,d} fits into the vector bundle diagram

(7.9)

Υ̃(Θ, P, P′) ⟶^{ρ_{P,P′}^{Ψ,d}} Υ(Θ, P)
↓  ↓
O^{asd}(Θ, P, P′, δ) ⟶^{ρ_{P,P′}^{Θ,d}} O^{asd}(Θ, P)

The map ρ_{P,P′}^{Ψ,d} is not injective because of the action of the symmetric group, but this will not affect our construction.

We also define a map

(7.10)  ρ_{P,P′}^{Ψ,u} : Υ̃(Θ, P, P′) → Υ(Θ, P′)

which fits into the vector bundle diagram

(7.11)

Υ̃(Θ, P, P′) ⟶^{ρ_{P,P′}^{Ψ,u}} Υ(Θ, P′)
↓  ↓
O^{asd}(Θ, P, P′, δ) ⟶^{ρ_{P,P′}^{Θ,u}} O^{asd}(Θ, P′)

Then, the following lemma, analogous to Lemma 5.11, is the key to the construction of the bundle Υ_{spl,κ}^{i}.

Lemma 7.2. Let P < P′ be partitions of N_{κ} with |P| > 1. Let Υ̃(Θ, P, [P′]) be the space defined in (7.8). Then, the following diagram of vector bundles,

(7.12)

Υ̃(Θ, P, P′) ⟶^{ρ_{P,P′}^{Ψ,d}} Υ(Θ, P)
ρ_{P,P′}^{Ψ,u} ↓  ^{φ′_{Θ,P}} ↓
Υ(Θ, P′) ⟶^{φ′_{Θ,P′}}  𝔙̄_{κ}

covers the diagram (5.40), is Spin^{u}(4) equivariant, and commutes.

Proof. That the diagram (7.12) covers the diagram (5.40) follows from the definition of the maps ρ_{P,P′}^{Ψ,d} and ρ_{P,P′}^{Ψ,u} (see the diagrams (7.9) and (7.11)) and the fact that the of the spicing maps φ′_{Θ,P} cover the splicing maps γ′_{Θ,P}.

The Spin^{u}(4) equivariance of the diagram follows from that of the splicing maps φ′_{Θ,P}. The commutativity of the diagram follows from the definition of the splicing maps φ′_{Θ,P}

and the property (5.27) which appears, in (5.35), in the definition of the overlap space Õ(Θ, P, P′, δ).

In detail, assume that the point in Υ̃(Θ, P, P′) is given by data consisting of points x_{P} ∈ Δ^{∘}(Z_{κ}(δ), P) (for P ∈ P), points y_{Q} ∈ Δ^{∘}(Z_{|P |}(δ_{P}), P′_{P}) (for Q ∈ P′_{P}), and [A_{Q}, F_{Q}^{s}, v_{Q}, Ψ_{Q}] ∈ Ῡ_{spl,|Q|}. Then, the first map in the composition φ′_{Θ,P} ∘ ρ_{P,P′}^{Ψ,d} entails defining sections Ψ_{P} (for P ∈ P) through the splicing maps φ_{Θ,P′} in the definition of the map ρ_{P,P′}^{Ψ,d} by

Ψ_{P} = ∑_{Q∈P′_{P}} χ_{y_{Q},λ_{Q}}c_{y_{Q},1}^{∗}Ψ_{Q},

where λ_{Q} = λ([A_{Q}, F_{Q}^{s}, v_{Q}]). Applying the splicing map φ′_{Θ,P} to the sections Ψ_{P} will give, for

λ_{P} = λ (γ′_{Θ,P′_{P}} ((y_{Q}, [A_{Q}, F_{Q}^{s}, v_{Q}])_{Q∈P′_{P}})) ,

the section

(7.13)  \begin{aligned}∑_{P ∈P} χ_{x_{P} ,λ_{P}} c_{x_{P} ,1}^{∗}Ψ_{P} & = ∑_{P ∈P} ∑_{Q∈P′_{P}} χ_{x_{P} ,λ_{P}} c_{x_{P} ,1}^{∗}χ_{y_{Q},λ_{Q}}c_{y_{Q},1}^{∗}Ψ_{Q} \\ & = ∑_{Q∈P′} χ_{x_{P} ,λ_{P}} χ_{x_{P} +y_{Q},λ_{Q}}c_{x_{Q}+y_{Q},1}^{∗}Ψ_{Q} \end{aligned}

where the last equality follows from Lemma 5.8. Recall that by the condition (5.35), the points x_{P} and y_{Q} in the definition of the subspace Õ(Θ, P, P′) will satisfy (see condition (5.27)),

B(x_{P} + y_{Q}, \frac{1}{4}λ_{Q}^{1/3}) ⋐ B(x_{P} , \frac{1}{8}λ_{P}^{1/2}).

Because χ_{x,λ} is supported on B(x, \frac{1}{4}λ^{1/3}) and equal to one on B(x, \frac{1}{8}λ^{1/2}). the preceding inclusion implies that χ_{x_{P} ,λ_{P}} is equal to one on the support of χ_{x_{P} +y_{Q},λ_{Q}}, so the section in (7.13) is equal to

∑_{Q∈P′} χ_{x_{P} +y_{Q},λ_{Q}}c_{x_{Q}+y_{Q},1}^{∗}Ψ_{Q}.

This is exactly the section resulting from the applying the map φ′_{Θ,P′} ∘ ρ_{P,P′}^{Ψ,u} to the same data, proving the diagram (7.12) commutes. □

Proof of Proposition 7.1. The proof of this proposition is largely identical to the proof of Theorem 5.1 so we omit many of the details.

The induction begins with setting Υ_{spl,1}^{i} equal to the bundle Index(D_{1}^{∗}) on the top stratum of M̄_{spl,1}^{s,♮}(δ) = (0, δ) × SO(3) (which is given by)

(0, δ) × SU(2) ×_{±1} ℂ → (0, δ) × SO(3).

The bundle extends as a pseudo-vector bundle over the Uhlenbeck compactification, c(SO(3)) as

c (SU(2) ×_{±1} ℂ) → c(SO(3)).

Assuming that Ῡ_{spl,κ′}^{i} has been constructed for all κ′ < κ, we define a pseudo-vector bundle Ῡ^{i}(W_{κ}) → W_{κ} over the spliced end by the image of the splicing maps φ′_{Θ,P} on the domains in (7.7). This space is a pseudo-vector bundle because the overlap of the images of the maps φ′_{Θ,P} is controlled by Lemma 7.2.

The final property in Proposition 7.1 on the surjectivity of L^{2}-orthogonal projection follows from the results of [9, §8] (see also [3, Proposition 7.1.32]). We can then define an isotopy R_{D} of Ῡ^{i}(W_{κ}) by, for t ∈ [\frac{1}{2}, \frac{3}{4}],

R_{D} ([A, F^{s}, x, Ψ], t) = (R([A, F^{s}, x, Ψ], t), (1 − 4(t − \frac{1}{2}))Ψ + 4(t − \frac{1}{2})Π_{A+a(A)}Ψ)

where R is the isotopy defined in (5.76) and for G([A, F^{s}, x] = [A+a(A), F^{s}, x] is the gluing map from (5.75), Π_{A+a(A)} denotes L^{2}-orthogonal onto Coker D_{A+a(A)}. For for t ∈ (−∞, \frac{1}{2}], the isotopy R_{D}(·, t) is equal to the identity and for t ∈ [\frac{3}{4}, 1], the isotopy R_{D}(·, t) is defined by pulling back the section Ψ by the centering map.

The bundle Ῡ_{spl,κ}^{i} is then defined by the union of the image Ῡ^{i}(W_{κ}) under the map R_{D}(·, t(·)) where t is the function defined in (5.77) with Index(D_{κ}^{∗}) restricted to the complement of

(M̄_{κ}^{s,♮}(δ) − R(1, W_{κ}))

as in the definition of M̄_{spl,κ}^{s,♮}(δ) in the proof of Theorem 5.1. □

8. The obstruction bundle

The gluing map is an embedding of M̄_{𝔱,𝔰}^{vir} into C̄_{𝔱} defined by a perturbation of the image of the splicing map. The intersection of the image of the gluing map with M̄_{𝔱} is defined by the zero-locus of a finite-dimensional pseudo-bundle, the obstruction bundle, which is a smooth vector bundle on each stratum of M̄_{𝔱,𝔰}^{vir}. In this section, we define this obstruction bundle.

The obstruction bundle is the direct sum of the Seiberg-Witten obstruction bundle, defined in §8.2, and the instanton obstruction pseudo-bundle, defined in §8.3. The gluing map, defined in Theorem 8.2, is homotopic to the global splicing map and the image of the gluing map contains a neighborhood of M_{𝔰} × Sym^{ℓ}(X) in M̄_{𝔱}. The intersection of the image of the gluing map with M̄_{𝔱} is given by the zero-locus of a section of the obstruction bundles as discussed in §8.4.

8.1. The infinite-dimensional obstruction bundle. The infinite-dimensional obstruction bundle in which the monopole map 𝔖 takes values is given by

(8.1)  𝔙_{𝔱} = C̃_{𝔱} ×_{G_{𝔱}} L_{k−1}^{2}(Λ^{+} ⊗ 𝔤_{𝔱}) ⊕ L_{k−1}^{2}(V^{−}) → C_{𝔱}.

The S^{1} action on C_{𝔱} lifts to an S^{1} action on the bundle (8.1) given by scalar multiplication on the element of L_{k−1}^{2}(V^{−}).

We extend the above bundle to a pseudo-bundle over C̄_{𝔱} by

(8.2)  𝔙̄_{𝔱} = ⊔_{ℓ} (𝔙_{𝔱(ℓ)} × Sym^{ℓ}(X)) → C̄_{𝔱} = ⊔_{ℓ} (C_{𝔱(ℓ)} × Sym^{ℓ}(X)) .

The monopole map is a continuous map 𝔖 : C̄_{𝔱} → 𝔙̄_{𝔱}.

8.2. The Seiberg-Witten obstruction. Recall from [9] or [11, §3.6.1] that the background or Seiberg-Witten component of the obstruction is constructed by splicing in sections of the obstruction bundle

π_{N}^{∗}Ξ_{𝔱,𝔰} → N_{𝔱,𝔰}(δ),

where π_{N} : N_{𝔱,𝔰}(δ) → M_{𝔰} and Ξ_{𝔱,𝔰} ≅ M_{𝔰}×ℂ^{r_{Ξ}} is a trivial bundle. The embedding N_{𝔱,𝔰}(δ) → C_{𝔱(ℓ)} is covered by an vector bundle embedding:

π_{N}^{∗}Ξ_{𝔱,𝔰} ⟶ 𝔙_{𝔱(ℓ)}
↓  ↓
N_{𝔱,𝔰}(δ) ⟶ C_{𝔱(ℓ)}

The above vector bundle map is S^{1} equivariant with respect to the actions described in [11, §3.6.1] and is defined by the gauge group quotient of a vector bundle embedding

π_{N}^{∗} Ξ̃_{𝔱,𝔰} → C̃_{𝔱(ℓ)} × L_{k−1}^{2}(Λ^{+} ⊗ 𝔤_{𝔱(ℓ)} ⊕ V_{𝔱(ℓ)}^{−}),

where Ξ̃_{𝔱,𝔰} → M̃_{𝔰} satisfies Ξ̃_{𝔱,𝔰}/G_{𝔰} = Ξ_{𝔱,𝔰}.

Define

Ξ_{𝔰}(𝔱, P) = π_{N}^{∗} Ξ̃_{𝔱,𝔰} ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P}) ⊂ π_{N}^{∗} Ξ̃_{𝔱,𝔰} ×_{G_{𝔰}} Gl(𝔱, 𝔰, P).

Then, the crude splicing construction from §6.3 (multiplying the obstruction sections by the cut-off functions used to define the flattened section Φ′ in the definition of the flattening map Θ_{P}) defines a vector bundle embedding,

φ′′_{𝔰}(P) : Ξ_{𝔰}(𝔱, P) → 𝔙̄_{𝔱},

covering the crude splicing map:

π_{N}^{∗} Ξ̃_{𝔱,𝔰} ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P}) ⟶^{φ′′_{𝔰}(P)} 𝔙̄_{𝔱}
↓  ↓
Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P, g_{P}) ⟶^{γ′′_{𝔱,𝔰,P}} C̄_{𝔱}

The crude obstruction splicing maps φ′′_{𝔰}(P′) are invariant under the action of the symmetric group and thus, for P < P′ define a crude obstruction splicing map

φ′′_{𝔰}([P < P′]) : π_{N}^{∗} Ξ̃_{𝔰} ×_{G_{𝔰}} O(𝔱, 𝔰, [P < P′]) → 𝔙̄_{𝔱}

covering the crude splicing map γ′′_{𝔱,𝔰,[P<P′]} defined in (6.38).

We can control the overlaps of the images of the embeddings φ′′_{𝔰}(P) exactly as was done in the construction of the space of global splicing data. The overlap maps, ρ_{P,[P′]}^{N,𝔱,𝔰,u} and

id_{N} × ρ_{P,[P′]}^{𝔱,𝔰,d}, appearing in the diagram (6.40) are covered by obvious bundle maps,

(8.3)

ρ_{P,[P′]}^{Ξ,𝔱,𝔰,u} : π_{N}^{∗} Ξ̃_{𝔰} ×_{G_{𝔰}} O(𝔱, 𝔰, P, [P′]) → π_{N}^{∗} Ξ̃_{𝔰} ×_{G_{𝔰}} O(𝔱, 𝔰, [P < P′])
id_{Ξ} × ρ_{P,[P′]}^{𝔱,𝔰,d} : π_{N}^{∗} Ξ̃_{𝔰} ×_{G_{𝔰}} O(𝔱, 𝔰, P, [P′]) → π_{N}^{∗} Ξ̃_{𝔰} ×_{G_{𝔰}} O(𝔱, 𝔰, P).

Then, exactly as in the proof of Proposition 6.9 (specifically the equality of the section Φ′_{0}), the diagram

(8.4)  \begin{matrix}π_{N}^{∗} Ξ̃_{𝔰} ×_{G_{𝔰}} O(𝔱, 𝔰, P, [P′]) & ⟶^{ρ_{P,[P′]}^{Ξ,𝔱,𝔰,u}} & π_{N}^{∗} Ξ̃_{𝔰} ×_{G_{𝔰}} O(𝔱, 𝔰, [P < P′]) \\ ^{id_{Ξ}×ρ_{P,[P′]}^{𝔱,𝔰,d}}↓ & & ^{φ′′_{𝔰}([P<P′])}↓ \\ π_{N}^{∗} Ξ̃_{𝔰} ×_{G_{𝔰}} O(𝔱, 𝔰, P) & ⟶^{φ′′_{𝔰}(P)} & 𝔙̄_{𝔱} \end{matrix}

commutes and covers the diagram (6.40).

We can therefore define a bundle

(8.5)  Ῡ_{𝔱,𝔰}^{s} → M̄_{𝔱,𝔰}^{vir}

as the union of the bundles Ξ_{s}(𝔱, P) as P varies over the partitions P of N_{ℓ}, patched together on the overlaps by the diagram (8.4). The embeddings φ′′_{s}(P) fit together to define a vector bundle embedding

φ′′_{s} : Ῡ_{𝔱,𝔰}^{s} → M̄_{𝔱,𝔰}^{vir}

covering the embedding given by the crude splicing maps.

The embedding φ′′_{s} suffers from the same failings as the crude splicing map: it is not equal to the identity. We therefore define a new vector bundle embedding following the isotopies of (6.55). That is, let φ′_{s}(P) be the standard slicing map

φ′_{s}(P) : Ξ_{s}(𝔱, 𝔰, P) → 𝔙̄_{𝔱}

defined by using the cut-off functions vanishing on balls B(x_{P} , 4λ_{P}^{1/3}). Define a sequence of vector bundle embeddings

φ′′_{s,j} : Ῡ_{𝔱,𝔰}^{s} → 𝔙̄_{𝔱}

exactly the embeddings γ′′_{M,j} were defined in (6.55), with the map φ′_{s}(P) in place of γ′_{𝔱,𝔰,P}. Define

(8.6)  φ′_{s}(𝔱, 𝔰) : Ῡ_{𝔱,𝔰}^{s} → 𝔙̄_{𝔱}

to be the end of the sequence of vector bundle embeddings φ′′_{s,j}. It is a vector bundle embedding, covering the global splicing map γ′_{M} of (6.56).

8.3. The instanton obstruction. The other component of the obstruction is given by splicing in the spliced-ends obstruction bundle over M̄_{spl,κ}^{s,♮}(δ).

8.3.1. The frame bundles. In [11, §3.2], we defined a frame bundle

Fr_{ℂℓ(T^{∗}X)}(V) → Fr(T X) → X

which is a U(2) bundle over Fr(T X) and a Spin^{u}(4) bundle over X. The fiber over F ∈ Fr(T X)|_{x} is given by the Clifford module isomorphisms from the standard Clifford module Δ ⊗ ℂ^{2} to V |_{x} with respect to the isomorphism ℝ^{4} ≅ T^{∗}X|_{x} defined by F. The homomorphisms from Spin^{u}(4) → SO(3) × SO(4) define a bundle map

Fr_{ℂℓ(T^{∗}X)}(V) → Fr(𝔤_{𝔱}) ×_{X} Fr(T X).

Analogously to the definition of the gluing data bundle Fr(𝔱, 𝔰, P, g_{P}) in (6.2), we define

(8.7)  Fr(V, P, g_{P}) → Fr(𝔱, 𝔰, P, g_{P}),

by

Fr(V, P, g_{P}) = {(F_{1}^{V} , …, F_{ℓ}^{V}) ∈ ∏_{i=1}^{ℓ} Fr_{ℂℓ(T^{∗})}(V_{𝔱(ℓ)})|_{Δ^{∘}(X^{ℓ},P)} :
F_{i}^{V} = F_{j}^{V} if and only there is P ∈ P with i, j ∈ P }.

The structure group of Fr(V, P, g_{P}) → Δ^{∘}(X^{ℓ}, P) is

(8.8) G̃^{V} (P) = {(G_{1}, …, G_{ℓ}) ∈ ∏_{i=1}^{ℓ} Spin^{u}(4) :

G_{i} = G_{j} if there is P ∈ P with i, j ∈ P }.

The structure group of the bundle Fr(V, P, g_{P}) → Σ(X^{ℓ}, P) is

(8.9)  G^{V} (P) = G̃^{V} (P) ⋉ 𝔖(P).

We then define the instanton obstruction bundle for the partition P by

(8.10)  Ob(𝔱, 𝔰, P) = Fr(V, P, g_{P}) ×_{G^{V} (P)} ∏_{P ∈P} D̄_{spl,|P |}^{s,♮}.

8.3.2. The splicing map. For each partition, P, of N_{ℓ}, we define a splicing map for the instanton obstruction,

(8.11)  φ_{𝔱,𝔰,P} : Ñ(δ) ×_{G_{𝔰}} Ob(𝔱, 𝔰, P) → 𝔙̄_{𝔱},

which will cover the crude splicing map γ′′_{𝔱,𝔰,P}. Frames (F_{P}^{V})_{P ∈P} ∈ Fr(V, P)|_{y}, where y = (y_{P})_{P ∈P} and a spin^{u} connection A determine embeddings

(8.12)  ϕ(A, F_{P}^{V}) : V_{κ_{i}}|_{B(0,8λ^{1/3})} ≅ V_{𝔱}|_{B(y_{P} ,8λ^{1/3})}

as described in [11, §3.4.1]. If χ is a cut-off function supported on B(0, 8λ^{1/3}) and Ψ is a section of V_{κ_{i}}, then we write ϕ(A, F^{V})(χΨ) for the section of V_{𝔱}|_{B(x,8λ^{1/3})} given by the isomorphism (8.12).

To ensure the images of the different splicing maps fit together coherently, we introduce a flattening map on spin^{u} connections as was done for metrics and connections in §6.2. That is, given a spin^{u} connection A_{0} on V_{𝔱(ℓ)} we follow the construction of Lemma 6.4 and for each y ∈ Σ(X^{ℓ}, P) define a locally flattened spin^{u} connection Θ^{u}(A, y) on V_{𝔱(ℓ)} satisfying

(1) For y = (y_{P})_{P ∈P}, Θ^{u}(A_{0}, y) is flat on

∪_{P ∈P}B(y_{p}, 4s̃_{P} (y)^{1/3})

(2) If P < P′ and y ∈ Σ(X^{ℓ}, P) and y′ ∈ Σ(X^{ℓ}, P′)∩U(X^{ℓ}, P) satisfy π(X^{ℓ}, P)(y′) = y,

then

Θ^{u}(A_{0}, y) = Θ^{u}(A_{0}, y′).

Henceforth, ϕ(A_{0}, F^{V}) will denote the isomorphism obtained by the locally flattened connection Θ^{u}(A_{0}, y) where F^{V} lies over the support of y.

For (A_{0}, Φ_{0}) ∈ N(δ) ⊂ C_{𝔱(ℓ)} and a point in Ob(𝔱, 𝔰, P) given by the data (F_{P}^{V})_{P ∈P} ∈ Fr(V, P, g_{P}), [A_{P} , F_{P}^{s} , v_{P}] ∈ M̄_{spl,|P |}^{s,♮}(δ_{P}) [A_{P} , F_{P}^{s} , v_{P} , Ψ] ∈ Ῡ_{spl,|P |}, and

γ′′_{𝔱,𝔰,P}((A_{0}, Φ_{0}), F_{P}^{V} , [A_{P} , F_{P}^{s} , v_{P}])_{P ∈P} = (A′′, Φ′′, x),

the instanton obstruction splicing map is then defined by (8.13)

φ_{𝔱,𝔰,P} ((A_{0}, Φ_{0}), (F_{P}^{V} , [A_{P} , F_{p}^{s}, v_{P} , Ψ_{P}])_{P ∈P}) = (A′′, Φ′′, x, ∑_{P ∈P} ϕ(A_{0}, F_{P}^{V})(χ_{0,λ_{P}} Ψ_{P}))

where the cut-off function χ_{0,λ_{P}} vanishes on B(0, \frac{1}{8}λ_{P}^{1/3}) and is equal to one on B(0, \frac{1}{4}λ_{P}^{1/2}).

8.3.3. The overlap space and maps. As is now standard, to control the overlap of the images of the splicing maps φ_{𝔱,𝔰,P} and φ_{𝔱,𝔰,P′}, we introduce a space of overlap data and overlap maps. Analogously to the definition of the overlap space Gl(𝔱, 𝔰, P, [P′]) in (6.22), we define

Ob(𝔱, 𝔰, P, [P′]) → Gl(𝔱, 𝔰, P, [P′])

by

(8.14) \begin{matrix}Ob(𝔱, 𝔰, P, [P′]) \\ = Fr(V, P, g_{P}) ×_{G^{V} (P)} ∐_{P′′∈[P<P′]} ∏_{P ∈P} (Δ^{∘}(Z_{|P |}(δ_{P}), P′′_{P}) × ∏_{Q∈P′′_{P}} Ῡ_{spl,|Q|}) . \end{matrix}

We define overlap maps,

ρ_{P,P′}^{V,d} : Ob(𝔱, 𝔰, P, [P′]) → Ob(𝔱, 𝔰, P)

covering the overlap maps ρ_{P,[P′]}^{𝔱,𝔰,d} by applying the obstruction splicing map φ_{Θ,P′′_{P}} defined in (7.5) to the data given by the factor

Δ^{∘}(Z_{|P |}(δ_{P}), P′′_{P}) × ∏_{Q∈P′′_{P}} Ῡ_{spl,|Q|}

in the definition of Ob(𝔱, 𝔰, P, [P′]). More formally, the downwards overlap map is defined by

(8.15)  ρ_{P,P′}^{V,d} = id_{Fr(V)} × ∐_{P′′∈[P<P′]} ∏_{P ∈P} φ_{Θ,P′′_{P}}

where id_{Fr(V)} is the identity map on Fr(V, P, g_{P}).

We define an upwards overlap map to the space,

Ob(𝔱, 𝔰, [P < P′]) = ( ∐_{P′′∈[P<P′]} Ob(𝔱, 𝔰, P′′)) /𝔖(P)

exactly as was done in (6.31) and (6.33). Recall that the map ρ_{P,[P′]}^{𝔱,𝔰,u} was defined by a parallel translation of the frames in the domain O(𝔱, 𝔰, P, [P′], g_{P}) and leaving the S^{4} connections unchanged. Note that this parallel translation is done with respect to the locally flattened connection Θ^{u}(A_{0}, y). Thus, the map

(8.16)  ρ_{P,[P′]}^{N,V,u} : Ñ(δ) ×_{G_{𝔰}} Ob(𝔱, 𝔰, P, [P′]) → Ñ(δ) ×_{G_{𝔰}} Ob(𝔱, 𝔰, [P < P′])

can be defined identically, using the flattened spin^{u} connection A_{0} to parallel translate the frames of Fr_{ℂℓ(T^{∗}X)}(V_{𝔱(ℓ)}) and leaving the elements of Ῡ_{spl,|Q|} unchanged.

By the invariance of the obstruction splicing maps under the action of the symmetric group, the splicing maps φ_{𝔱,𝔰,P′′} for P′′ ∈ [P < P′] define a splicing map,

(8.17)  φ_{𝔱,𝔰,[P<P′]} : Ñ(δ) ×_{G_{𝔰}} Ob(𝔱, 𝔰, [P < P′]) → 𝔙̄_{𝔱}

which covers the crude splicing map γ′′_{𝔱,𝔰,[P<P′]} of (6.38). We then have the following relation between the splicing maps φ_{𝔱,𝔰,P} and φ_{𝔱,𝔰,[P<P′]}. Lemma 8.1. Let P < P′ be partitions of N_{ℓ}. Then the following diagram commutes:

(8.18)

Ñ(δ) ×_{G_{𝔰}} Ob(𝔱, 𝔰, P, [P′]) ⟶^{ρ_{P,[P′]}^{N,V,u}} Ñ(δ) ×_{G_{𝔰}} Ob(𝔱, 𝔰, [P < P′])
id_{N} ×ρ_{P,[P′]}^{V,d}↓  ^{φ_{𝔱,𝔰,[P<P′]}} ↓
Ñ(δ) ×_{G_{𝔰}} Ob(𝔱, 𝔰, P) ⟶^{φ_{𝔱,𝔰,P}}  𝔙̄_{𝔱}

where id_{N} is the identity map on Ñ(δ), covering the diagram (6.40).

Proof. That the diagram (8.18) covers the diagram (6.40) follows immediately from the definition of the overlap and obstruction splicing maps. Then, as in the proof of Proposition 6.9, assume that the splicing maps φ_{𝔱,𝔰,P} and φ_{𝔱,𝔰,P′} are defined at points y ∈ Σ(X^{ℓ}, P) and y′ ∈ Σ(X^{ℓ}, P′) respectively where π(X^{ℓ}, P)(y′) = y. Thus, for P′ ∈ P′ and P′ ⊆ P ∈ P, the inclusion

B(y_{P′}, 8s̃(y′)^{1/3}) ⋐ B(y_{P} , 4s̃(y)^{1/2})

holds by (6.19). Because the metric and spin^{u} connection used in the splicing argument are flat on the above balls, the lemma then follows from the argument giving Lemma 7.2. □

The commutativity of the diagram (8.18) and the definition of the space M̄_{𝔱,𝔰}^{vir} as the union of the spaces

Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P)

implies that we can define a vector pseudo-bundle

(8.19)  Ῡ_{𝔱,𝔰}^{i} → M̄_{𝔱,𝔰}^{vir}

as the union of the pseudo-bundles

Ñ(δ) ×_{G_{𝔰}} Ob(𝔱, 𝔰, P) → Ñ(δ) ×_{G_{𝔰}} O(𝔱, 𝔰, P),

with the overlaps identified by the diagram (8.18).

The splicing maps φ_{𝔱,𝔰,P} define a vector bundle embedding of Ῡ_{𝔱,𝔰}^{i} into 𝔙̄_{𝔱} which covers the embedding of M̄_{𝔱,𝔰}^{vir} into C̄_{𝔱} given by the crude splicing maps. Finally, we define an embedding

(8.20)  φ′_{i} : Ῡ_{𝔱,𝔰}^{i} → 𝔙̄_{𝔱}

covering the global splicing map γ′_{M} of (6.56) by replacing the triple (A′′, Φ′′, x) in the definition (8.13) with the triple (A′, Φ′, x) given by the global splicing map γ′_{M}.

8.4. The gluing theorem. We are now in a position to state the relevant gluing theorem. Theorem 8.2. There is a continuous, S^{1}-equivariant embedding

(8.21)  γ_{M} : M̄_{𝔱,𝔰}^{vir} → C̄_{𝔱}

which is homotopic through S^{1}-equivariant embeddings to the global splicing map γ′_{M}, smooth on each stratum of M̄_{𝔱,𝔰}^{vir}, and equal to the identity on N(δ) × Sym^{ℓ}(X). In addition, there are continuous, S^{1}-equivariant sections 𝔬_{s} and 𝔬_{i} of the bundles Ῡ_{𝔱,𝔰}^{s} and Ῡ_{𝔱,𝔰}^{i} such that

(1) The section 𝔬 = 𝔬_{s} ⊕ 𝔬_{i} of Ῡ_{𝔱,𝔰}^{s} ⊕ Ῡ_{𝔱,𝔰}^{i} vanishes transversely on the top stratum of

M̄_{𝔱,𝔰}^{vir},

(2) The restriction of γ_{M} to the zero-locus 𝔬^{−1}(0) parameterizes a neighborhood of M_{𝔰}× Sym^{ℓ}(X) in M̄_{𝔱}.

9. Cohomology

In this section, we compute the pullback of the cohomology classes by the global splicing map. We define the cohomology classes and maps in terms of which these pullbacks will be expressed in §9.1. We compute the pullbacks of the cohomology classes μ_{p}(β) and μ_{c} in §9.2 the Euler class of the Seiberg-Witten component of the obstruction in §9.3. The computation of the Euler class of the instanton component of the obstruction occupies §9.4, where we compute the restriction of this Euler class to the top stratum of the subspaces U(𝔱, 𝔰, P_{i}) ⊂ M̄_{𝔱,𝔰}^{vir}/S^{1} and §9.5 where discuss how these local computations yield the global cohomology class. Finally, in §9.6, we define some extensions of these cohomology classes from the top stratum of M̄_{𝔱,𝔰}^{vir}/S^{1} the complement of the reducible monopoles.

9.1. Definitions.

9.1.1. Subspace and maps. We begin by defining subspaces of M̄_{𝔱,𝔰}^{vir}. First,

(9.1)  M_{𝔱,𝔰}^{vir} ⊂ M̄_{𝔱,𝔰}^{vir}

will denote the top stratum which is mapped to C_{𝔱} by the splicing and gluing maps γ′_{M} and γ_{M}. Let

(9.2)  ι : M_{𝔱,𝔰}^{vir} → M̄_{𝔱,𝔰}^{vir}

be the inclusion. Next, we define the complement of the reducible points,

(9.3)  M̄_{𝔱,𝔰}^{vir,∗} = M̄_{𝔱,𝔰}^{vir} − M_{𝔰} × Sym^{ℓ}(X).

The S^{1} action is free on the subspace M̄_{𝔱,𝔰}^{vir,∗}.

The map

π_{N} : M̄_{𝔱,𝔰}^{vir} → N_{𝔱_{0},𝔰}(δ),

was defined in (6.43) and we define

(9.4)  π_{𝔰} : M̄_{𝔱,𝔰}^{vir} → M_{𝔰}

as the composition of π_{N} with the projection N_{𝔱_{0},𝔰}(δ) → M_{𝔰}. Recall that the projection

π_{X} : M̄_{𝔱,𝔰}^{vir} → Sym^{ℓ}(X)

was defined in (6.57). We will also write

(9.5)  π_{𝔰,X} : M̄_{𝔱,𝔰}^{vir} → M_{𝔰} × Sym^{ℓ}(X)

and

(9.6)  π_{N,X} : M̄_{𝔱,𝔰}^{vir} → N_{𝔱_{0},𝔰}(δ) × Sym^{ℓ}(X)

for the projections.

We use the same notation to denote the projections from M̄_{𝔱,𝔰}^{vir}/S^{1} (although note that π_{N} maps this quotient to N_{𝔱_{0},𝔰}(δ)/S^{1}).

9.1.2. Cohomology classes. We define some cohomology classes on M̄_{𝔱,𝔰}^{vir,∗}/S^{1}.

Definition 9.1. Let ν ∈ H^{2}(M̄_{𝔱,𝔰}^{vir,∗}/S^{1}; ℤ) be the first Chern class of the S^{1} bundle,

(9.7)  M̄_{𝔱,𝔰}^{vir,∗} → M̄_{𝔱,𝔰}^{vir,∗}/S^{1}.

If

(9.8)  𝕃_{ν} = M̄_{𝔱,𝔰}^{vir,∗} ×_{S^{1}} ℂ,

to be the complex line bundle associated to the S^{1} action on M̄_{𝔱,𝔰}^{vir,∗} so c_{1}(𝕃_{ν}) = ν.

Definition 9.2. For β ∈ H_{•}(X; ℝ), let μ_{𝔰}(β) ∈ H^{2}(M_{𝔰}; ℤ) be the μ-class defined in §2. We will use the same notation for the pullback, by the projection π_{𝔰}, of these classes to M̄_{𝔱,𝔰}^{vir}/S^{1}. Define

(9.9)  𝕃_{𝔰} = M̃_{𝔰} ×_{G_{𝔰}} X × ℂ → M_{𝔰} × X.

Definition 9.3. If β ∈ H^{•}(X; ℝ) and π_{i} : X^{ℓ} → X is projection onto the i-th factor, let S^{ℓ}(β) ∈ H^{•}(Sym^{ℓ}(X); ℝ) be the cohomology class which pulls back to ∑_{i} π_{i}^{∗}β under the projection X^{ℓ} → Sym^{ℓ}(X).

The incidence locus,

(9.10)  V_{ℓ}(Δ) ⊂ Sym^{ℓ}(X) × X

is defined to be the points (x, y) ∈ Sym^{ℓ}(X) × X such that y is in the support of x. Alternately, one can describe it

V_{ℓ}(Δ) = (∪_{i}(π_{i} × id_{X})^{−1}(Δ)) /𝔖_{ℓ},

where π_{i} : X^{ℓ} → X is projection onto the i-th factor and Δ ⊂ X × X is the diagonal. Similarly, we define

(9.11)  V_{ℓ}(ν(Δ)) = (∪_{i}(π_{i} × id_{X})^{−1}(ν(Δ))) /𝔖_{ℓ},

where ν(Δ) ⊂ X × X is the normal bundle of Δ. Let

(9.12)  PD[V] ∈ H^{4}(Sym^{ℓ}(X) × X; ℤ),

be the image of the Thom class of the disk bundle V_{ℓ}(ν(Δ)) → V_{ℓ}(Δ) under the excision isomorphism and the map

H^{4}(Sym^{ℓ}(X) × X, Sym^{ℓ}(X) × X − V_{ℓ}(Δ); ℤ) → H^{4}(Sym^{ℓ}(X) × X; ℤ).

From the equality on Poincare duality given in [18, Theorem 30.6], we have Lemma 9.4. For any β ∈ H_{•}(X; ℝ),

PD[V]/β = S^{ℓ}(PD[β]).

9.2. The μ-classes. We describe the pullbacks of the cohomology classes μ_{c} and μ_{p}(β) by the gluing map in terms of the cohomology classes introduced in §9.1.

The computation of the pullback of μ_{c} is identical to that appearing in [11, Lemma 4.2].

Lemma 9.5. If μ_{c} ∈ H^{2}(C_{𝔱}^{∗,0}/S^{1}; ℤ) is the cohomology class defined in (2.44), γ_{M} is the gluing map of (8.21), and ι is the inclusion (9.2) then

(9.13)  γ_{M}^{∗}μ_{c} = −ι^{∗}ν.

To compute the pullback of the μ_{p} classes, we must study the incidence locus V_{ℓ}(Δ) more carefully. Define

(9.14)  O = M_{𝔱,𝔰}^{vir}/S^{1} × X − (π_{X} × id_{X})^{−1}V_{ℓ}(ν(Δ))

We will consider compare the pullback of the universal bundle (γ × id_{X})^{∗}𝔽_{𝔱}, restricted to O, with the same restriction of the bundle

(9.15)  𝔽_{𝔱,𝔰}^{vir} = ∪_{P} (Ñ(δ) × Gl(𝔱, 𝔰, P) ×_{G_{𝔰}×S^{1}} 𝔤_{𝔱})

which is defined analogously to the bundle in [11, Equation (4.7)]. The proof of [11, Lemma 4.7] then yields the following expression for p_{1}(𝔽_{𝔱,𝔰}^{vir}).

Lemma 9.6. Assume the spin^{u} structure 𝔱_{ℓ} admits a splitting 𝔱_{ℓ} = 𝔰 ⊕ 𝔰 ⊗ L. Let 𝔽_{𝔱,𝔰}^{vir} → M_{𝔱,𝔰}^{vir}/S^{1} ×X be the bundle defined in (9.15). Let π_{X,2} : M_{𝔱,𝔰}^{vir}/S^{1} ×X → X be the projection onto the second factor. Then,

(9.16)  p_{1}(𝔽_{𝔱,𝔰}^{vir}) = ι^{∗} ((2π_{𝔰} × id_{X})^{∗}c_{1}(𝕃_{𝔰}) − ν + π_{X,2}^{∗}c_{1}(L))^{2} .

The following comparison of 𝔽_{𝔱,𝔰}^{vir} with (γ_{M} × id_{X})^{∗}𝔽_{𝔱} is proven in [11, Lemma 4.3].

Lemma 9.7. Assume the spin^{u} structure 𝔱_{ℓ} admits a splitting 𝔱_{ℓ} = 𝔰 ⊕ 𝔰 ⊗ L. Let O ⊂ M_{𝔱,𝔰}^{vir}/S^{1} × X be the subspace defined in (9.14). Then,

(9.17)  (γ_{M} × id_{X})^{∗}𝔽_{𝔱}|_{O} ≅ 𝔽_{𝔱,𝔰}^{vir}|_{O}.

We specify the difference between the Pontrjagin classes of the bundles in Lemma 9.7 in the following.

Lemma 9.8. Assume the spin^{u} structure 𝔱_{ℓ} admits a splitting 𝔱_{ℓ} = 𝔰 ⊕ 𝔰 ⊗ L. Let PD[V] be the cohomology class defined in (9.12).

(9.18)  (γ_{M} × id_{X})^{∗}p_{1}(𝔽_{𝔱}) = p_{1}(𝔽_{𝔱,𝔰}^{vir}) − 4(ι × id_{X})^{∗}(π_{X} × id_{X})^{∗} PD[V].

Let γ_{1}, …, γ_{b} ∈ H_{1}(X; ℝ) be a basis with γ_{1}, …, γ_{b} ∈ H^{1}(X; ℝ) the basis satisfying ⟨γ_{i}^{∗}, γ_{j}⟩ = δ_{ij}. Let γ_{1}^{J,∗}, …, γ_{b}^{J,∗} ∈ H^{1}(M_{𝔰}; ℝ) be the related basis as defined in [15, Definition 2.2] (there written as r^{∗}γ_{i}^{J,∗}). Standard computations (compare [11, Lemma 4.10] and [16, Corollary 4.7]), Lemma 9.4, and the equality

c_{1}(𝕃_{𝔰}) = μ_{𝔰} × 1 + ∑_{i=1}^{b_{1}} γ_{i}^{J,∗} × γ_{i}^{∗}

from [15, Lemma 2.24] then yield the following.

Corollary 9.9. Assume the spin^{u} structure 𝔱_{ℓ} admits a splitting 𝔱_{ℓ} = 𝔰 ⊕ 𝔰 ⊗ L, so c_{1}(L) = c_{1}(𝔱)−c_{1}(𝔰). Let γ_{M} be the gluing map defined in (8.21) and ι the inclusion defined in (9.2). Then for x ∈ H_{0}(X; ℤ) a generator, γ_{i} ∈ H_{1}(X; ℤ) a generator as above, β ∈ H_{2}(X; ℝ), and [Y] ∈ H_{3}(X; ℝ), (9.19)

\begin{aligned}γ_{M}^{∗}μ_{c}(x) & = −ι^{∗}ν, \\ γ_{M}^{∗}μ_{p}(x) & = ι^{∗} (−\frac{1}{4}(2μ_{𝔰} − ν)^{2} + S^{ℓ}(x)) , \\ γ_{M}^{∗}μ_{p}(γ_{i}) & = ι^{∗} (−\frac{1}{2}(2μ_{𝔰} − ν)γ_{i}^{J,∗} + S^{ℓ}(γ_{i})) , \\ γ_{M}^{∗}μ_{p}(β) & = ι^{∗} (−\frac{1}{2}2μ_{𝔰} − ν)⟨c_{1}(𝔱) − c_{1}(𝔰), β⟩ − 2 ∑_{i<j} μ_{𝔰}(γ_{i}γ_{j})⟨γ_{i}^{∗} ⌣ γ_{j}^{∗}, β⟩ + S^{ℓ}(β)) , \\ γ_{M}^{∗}μ_{p}(β_{3}) & = ι^{∗} (− ∑_{i}(c_{1}(𝔱) − c_{1}(𝔰) ⌣ γ_{i}^{∗}, [Y]⟩μ_{𝔰}(γ_{i}) + S^{ℓ}(β)) , \end{aligned}

9.3. The Seiberg-Witten obstruction Euler class. The computation of the Euler class of the Seiberg-Witten, or background, component of the obstruction is identical to that given in [11, Lemma 4.11]. Lemma 9.10. Let r_{Ξ} be the complex rank of the background obstruction bundle Υ_{𝔱,𝔰}^{s} → M̄_{𝔱,𝔰}^{vir,∗}. Then,

(9.20)  e(Υ_{𝔱,𝔰}^{s}/S^{1}) = ι^{∗}(−ν)^{r_{Ξ}}.

9.4. The local instanton obstruction Euler class. We first give a description of the Euler class of the instanton component of the obstruction bundle, Υ_{𝔱,𝔰}^{i}/S^{1} → M_{𝔱,𝔰}^{vir}/S^{1} restricted to O(𝔱, 𝔰, P).

We first introduce an additional cohomology class. Let π_{𝔰,i} : M_{𝔰} × X^{ℓ} → M_{𝔰} × X be defined by the identity on M_{𝔰} and projection from X^{ℓ} onto the i-th factor of X. Then, the Chern classes of the bundle ⊕_{i=1}^{ℓ}π_{𝔰,i}^{∗}𝕃_{𝔰} are invariant under the action of the symmetric group and thus define cohomology classes

(9.21)  c_{𝔰,ℓ,i} ∈ H^{2i}(M_{𝔰} × Sym^{ℓ}(X); ℝ).

In addition, we will write

ι_{P} : Σ(X^{ℓ}, P) → Sym^{ℓ}(X), ι̃_{P} : Δ^{∘}(X^{ℓ}, P) → X^{ℓ},

for the inclusions. Finally, we use the generic term

(9.22)  c(𝔱) ∈ H^{•}(X; ℝ),

for rational characteristic classes of the bundle Fr_{ℂℓ(T^{∗}X)}(V) → X. These characteristic classes include c_{1}(𝔱), p_{1}(𝔱), p_{1}(X), and e(X). Proposition 9.11. The Euler class of the restriction of the obstruction bundle Υ_{𝔱,𝔰}^{i}/S^{1} to O(𝔱, 𝔰, P) is given by a polynomial in the cohomology classes

(9.23)  ν, π_{X}^{∗}ι_{P}^{∗}S^{ℓ}(c(𝔱)), and π_{X,𝔰}^{∗}ι_{P}^{∗}c_{𝔰,ℓ,i},

which is otherwise independent of X.

To prove Proposition 9.11, we begin by observing that the space O(𝔱, 𝔰, P) retracts to the intersection of O(𝔱, 𝔰, P) with the subspace

(9.24)  M̃_{𝔰} ×_{G_{𝔰}×S^{1}} Gl(𝔱, 𝔰, P).

We compute the pullback of the restriction of the Euler class of the instanton obstruction to the subspace (9.24) by the map,

(9.25)  M̃_{𝔰} ×_{G_{𝔰}×S^{1}} G̃l(𝔱, 𝔰, P) → M̃_{𝔰} ×_{G_{𝔰}×S^{1}} Gl(𝔱, 𝔰, P)

defined by the projection

G̃l(𝔱, 𝔰, P) = Fr(𝔱, 𝔰, P) ×_{G̃(P)} ∏_{P ∈P} M_{spl,|P |}^{s,♮}(δ_{P}) → Gl(𝔱, 𝔰, P),

where the group G̃(P) is defined in (6.3) (essentially by omitting the symmetric group component of G(P)). Observe that the preceding map appears in the diagram,

G̃l(𝔱, 𝔰, P) ⟶ Gl(𝔱, 𝔰, P)
↓  ↓
Δ^{∘}(X^{ℓ}, P) ⟶ Σ(X^{ℓ}, P)

For each P ∈ P, the projection G̃l(𝔱, 𝔰, P) → Δ^{∘}(X^{ℓ}, P) fits into a diagram

(9.26)

G̃l(𝔱, 𝔰, P) ⟶^{π_{G̃l,P}} Fr(𝔤_{0}) ×_{X} Fr(T X) ×_{SO(3)×SO(4)} M_{spl,|P |}^{s,♮}(δ_{P})
↓  ↓
Δ^{∘}(X^{ℓ}, P) ⟶^{π_{P}}  X

Then, we have the following: Lemma 9.12. The pullback of the restriction of Υ_{𝔱,𝔰}^{i}/S^{1} to O(𝔱, 𝔰, P) by the covering map (9.25) splits into a direct sum, ⊕_{P ∈P}π_{G̃l,P}^{∗} Υ_{𝔱,𝔰}^{i}(P)/S^{1}, where the bundle

(9.27)  Υ_{𝔱,𝔰}^{i}(P)/S^{1} → M̃_{𝔰} ×_{G_{𝔰}×S^{1}} Fr(𝔤_{ℓ}) ×_{X} Fr(T X) ×_{SO(3)×SO(4)} M_{spl,|P |}^{s,♮}(δ_{P})

is defined by

(9.28)

M̃_{𝔰} ×_{G_{𝔰}×S^{1}} Fr_{ℂℓ(T^{∗}X)}(V_{ℓ}) ×_{Spin^{u}(4)} Υ_{spl,|P |}^{i}
↓
M̃_{𝔰} ×_{G_{𝔰}×S^{1}} Fr(𝔤_{ℓ}) ×_{X} Fr(T X) ×_{SO(3)×SO(4)} M̄_{spl,|P |}^{s,♮}(δ_{P})

and the map π_{G̃l,P} is defined in the diagram (9.26).

The S^{1} action in (9.28) reflects the action of S^{1} on the infinite-dimensional obstruction space. Thus, following [11, Lemma 3.7], the S^{1} action on

(9.29)  M̃_{𝔰} ×_{G_{𝔰}} Fr_{ℂℓ(T^{∗}X)}(V) ×_{Spin^{u}(4)} Υ_{spl,|P |}^{i}

is given by the diagonal action of the action by ϱ_{𝔰}(e^{−iθ}) on Fr_{ℂℓ(T^{∗}X)}(V) and by scalar multiplication with weight one on the fibers of Υ_{spl,|P |}^{i}.

Lemma 9.12 implies that to compute the restriction of the instanton obstruction bundle to the subspace (9.24), it suffices to characterize the Euler class of the bundle (9.28). We use the notation

Gl(𝔱, 𝔰, P) = Fr(𝔤_{ℓ}) ×_{X} Fr(T X) ×_{SO(3)×SO(4)} M_{spl,|P |}^{s,♮}(δ_{P})

Then, the base of the bundle Υ_{𝔱,𝔰}^{i}(P)/S^{1} admits a product decomposition:

(9.30)  M̃_{𝔰} ×_{G_{𝔰}×S^{1}} Gl(𝔱, 𝔰, P) ≅ M_{𝔰} × S^{1}\ Gl(𝔱, 𝔰, P).

Define

(9.31) Υ_{X,P}^{i} /S^{1} = M_{𝔰} × S^{1}\ Fr_{ℂℓ(T^{∗}X)}(V_{ℓ}) ×_{Spin^{u}(4)} Υ_{spl,|P |}^{i} → M_{𝔰} × S^{1}\ Gl(𝔱, 𝔰, P).

As in the discussion of the S^{1} action on (9.29), the S^{1} action in (9.31) is given by the diagonal action of the action by ϱ_{𝔰}(e^{−iθ}) on Fr_{ℂℓ(T^{∗}X)}(V) and by scalar multiplication on the fibers of Υ_{spl,|P |}^{i}.

Lemma 9.13. Let 𝕃_{𝔰} → M_{𝔰} × X be the universal Seiberg-Witten line bundle defined in ??. Let π_{X,𝔰} : M_{𝔰} × S^{1}\ Gl(𝔱, 𝔰, P) → M_{𝔰} × X be the projection. Then,

(9.32)  Υ_{𝔱,𝔰}^{i}(P)/S^{1} ≅ π_{X,𝔰}^{∗}𝕃_{𝔰}^{∗} ⊗ Υ_{X,P}^{i} /S^{1}.

Proof. By the technical result [15, Lemma 3.27], the tensor product in (9.32) can be described as:

(9.33)  ((M̃_{𝔰} ×_{G_{𝔰}} X × S^{1})_{M_{𝔰}×X} Υ_{X,P}^{i} /S^{1}) /S^{1},

where S^{1} acts diagonally on the factor of S^{1} and on the fiber of Υ_{X,P}^{i} /S^{1}. Because the S^{1} quotient in the definition of Υ_{X,P}^{i} /S^{1} acts anti-diagonally (see the remarks following (9.31)), we can then say that the final S^{1} quotient in (9.33) acts diagonally on the factor of S^{1} and by the action ϱ_{𝔰}(e^{iθ}) on the factor of Fr_{ℂℓ(T^{∗}X)}(V) in the definition of Υ_{X,P}^{i} /S^{1}. One then defines a bundle map from the fibered product (9.33) to the bundle Υ_{𝔱,𝔰}^{i}(P)/S^{1} by

((A_{0}, Φ_{0}, x, e^{iθ}, F̃_{u}, [A, F_{s}, Ψ]) → ((A_{0}, Φ_{0}, e^{−iθ}F̃_{u}, [A, F_{s}, Ψ]) ,

where (A_{0}, Φ_{0}) ∈ M̃_{𝔰}, x ∈ X, e^{iθ} ∈ S^{1}, F̃_{u} ∈ Fr_{ℂℓ(T^{∗}X)}(V)|_{x}, and [A, F_{s}, Ψ] ∈ Υ_{spl,|P |}^{i}. Observe that the map is well-defined because the diagonal S^{1} action described earlier in the proof vanishes in the product e^{−iθ}F̃_{u}. The equivariance of this map with respect to the G_{𝔰} action is correct because the action of G_{𝔰} defining 𝕃_{𝔰} is given by:

(u, (A_{0}, Φ_{0}, x, z)) → (u_{∗}(A_{0}, Φ_{0}), x, u(x)^{−1}z) ,

and thus using e^{−iθ}F̃_{u} takes the preceding action of G_{𝔰} to the desired diagonal action on M̃_{𝔰} and Fr_{ℂℓ(T^{∗}X)}(V). □

Next, we characterize the (rational) Euler class of Υ_{X,P}^{i} /S^{1}. To this end, we introduce the cohomology class,

(9.34)  ν_{Gl} ∈ H^{2}(M_{𝔰} × S^{1}\ Gl(𝔱, 𝔰, P)),

(denoted ν_{𝔱} in [11, Definition 5.22]) to be the first Chern class of the S^{1} bundle,

(9.35)  M_{𝔰} × Gl(𝔱, 𝔰, P) → M_{𝔰} × S^{1}\ Gl(𝔱, 𝔰, P).

In [11, Lemma 5.23], the equation

(9.36)  ν = ν_{Gl} + 2π_{X,𝔰}^{∗}c_{1}(𝕃_{𝔰})

is proven. The advantage of the bundle (9.34) over π_{X,𝔰}^{∗}𝕃_{𝔰} is that the former pulls back by the projection M_{𝔰} × §^{1}\ Gl(𝔱, 𝔰, P) → S^{1}\ Gl(𝔱, 𝔰, P). Lemma 9.14. The rational Euler class,

e(Υ_{X,P}^{i} /S^{1}) ∈ H^{2|P |}(M_{𝔰} × S^{1}\ Gl(𝔱, 𝔰, P); ℝ),

is given by a polynomial in π_{X}^{∗}c(𝔱) and ν_{Gl} which is otherwise independent of X.

Proof. Observe that the bundle Υ_{X,P}^{i} /S^{1} is the pullback by the projection

M_{𝔰} × S^{1}\ Gl(𝔱, 𝔰, P) → S^{1}\ Gl(𝔱, 𝔰, P)

of the bundle

(9.37)  S^{1}\ Fr_{ℂℓ(T^{∗}X)}(V_{ℓ}) ×_{Spin^{u}(4)} Υ_{spl,|P |}^{i} → S^{1}\ Gl(𝔱, 𝔰, P).

The lemma will then follow if we can that the Euler class of the bundle (9.37) satisfies the conclusion of the lemma.

The kernel of the projection

(9.38)  (Ad_{SO(3)}^{u}, Ad_{SO(4)}^{u}) : Spin^{u}(4) → SO(3) × SO(4)

is the central S^{1} in Spin^{u}(4). By the identity (see [11, Equation (3.14)])

Fr_{ℂℓ(T^{∗}X)} /S^{1} = Fr(𝔤_{ℓ}) ×_{X} Fr(T X).

(where the S^{1} is the kernel of the homomorphism (9.38)), we can rewrite the space Gl(𝔱, 𝔰, P) as

Gl(𝔱, 𝔰, P) = Fr_{ℂℓ(T^{∗}X)}(V_{ℓ}) ×_{Spin^{u}(4)} M_{spl,|P |}^{s,♮}(δ_{P})

where Spin^{u}(4) acts on M_{spl,|P |}^{s,♮}(δ_{P}) via the projection (9.38) and the action of SO(3)×SO(4)

on M_{spl,|P |}^{s,♮}(δ_{P}). Let ι̃_{u} : Fr_{ℂℓ(T^{∗}X)}(V_{ℓ}) → ESpin^{u}(4) be the classifying map, appearing in the diagram:

Fr_{ℂℓ(T^{∗}X)}(V_{ℓ}) ⟶^{ι̃_{u}} ESpin^{u}(4)
↓  ↓
X  ⟶^{ι_{u}} BSpin^{u}(4)

Recall from [1, §4], that the map [A] → [D_{A}] where D_{A} is the Dirac operator defined by the connection A defines a continuous map,

f_{κ} : B_{κ}^{s} → F_{κ},

where F_{κ} is the space of Fredholm operators of index κ. The space F_{κ} is homotopic to BU = lim_{n} BU(n). Although there is no index bundle defined for the Dirac operators parameterized by B_{κ}^{s} because it is not compact, the restriction of the pullback of the universal Chern classes in BU by f_{κ} to the subspace K_{κ}^{s} ⊂ B_{κ}^{s} where the cokernel of the Dirac operator vanishes are equal to the Chern class of the vector bundle defined by Index(D) on this subspace. By property (iv) in Proposition 7.1, the Chern classes of the bundle

Υ_{spl,|P |}^{i} → M_{spl,|P |}^{s,♮}

are given by the pullback of the universal Chern classes in H^{•}(BU) by the map f_{|P |}. By the Spin^{u}(4)-equivariance of the Dirac operator, the map f_{κ} extends to a map

F_{κ} : ESpin^{u}(4) ×_{Spin^{u}(4)} B_{κ}^{s} → F_{κ},

such that the restriction of F_{κ} to

ESpin^{u}(4) ×_{Spin^{u}(4)} K_{κ}^{s}

is the composition of a classifying map for the index bundle with the inclusion BU(κ) → BU. We conclude that the classifying map for the bundle

Υ_{X,P}^{i} = Fr_{ℂℓ(T^{∗}X)}(V_{ℓ}) ×_{Spin^{u}(4)} Υ_{spl,|P |}^{i} → Fr_{ℂℓ(T^{∗}X)}(V_{ℓ}) ×_{Spin^{u}(4)} M_{spl,|P |}^{s,♮}(δ_{P}),

factors through the composition

(9.39) Fr_{ℂℓ(T^{∗}X)}(V_{ℓ}) ×_{Spin^{u}(4)} M_{spl,|P |}^{s,♮}(δ_{P}) ⟶^{ι̃_{u}×ι_{M}} ESpin^{u}(4) ×_{Spin^{u}(4)} B_{|P |}^{s} ⟶^{F_{|P |}} F_{|P |}

where ι_{M} : M_{spl,|P |}^{s,♮}(δ_{P}) → B_{|P |}^{s} is the inclusion. The bundle map ι̃_{u} × ι_{M} descends to the S^{1} quotients:

ι̃_{u} × ι_{M} : S^{1}\ (Fr_{ℂℓ(T^{∗}X)}(V_{ℓ}) ×_{Spin^{u}(4)} M_{spl,|P |}^{s,♮}(δ_{P})) → S^{1}\ (ESpin^{u}(4) ×_{Spin^{u}(4)} B_{|P |}^{s})

Then, because the rational cohomology of B_{|P |}^{s} is trivial (see [3, Lemma 5.1.14]), the rational cohomology of the space

S^{1}\ (ESpin^{u}(4) ×_{Spin^{u}(4)} B_{|P |}^{s})

is generated by the first Chern class of the S^{1} action and cohomology classes pulled back by the projection

S^{1}\ (ESpin^{u}(4) ×_{Spin^{u}(4)} B_{|P |}^{s}) → BSpin^{u}(4).

Under the map ι̃_{u} × ι_{M}, the first Chern class of the S^{1} action pulls back to ν_{Gl} while the cohomology classes pulled back from BSpin^{u}(4) pullback to characteristic classes of Fr_{ℂℓ(T^{∗}X)}(V), which we are denoting by c(𝔱). The conclusion of the lemma then follows from the factoring of the classifying map for the bundle (9.37) given in (9.39). □

Remark 9.15. Unlike the case ℓ = 1, the action of G_{𝔰} on S^{1}\ Gl(𝔱, 𝔰, P) is not trivial because there can be more than one frame in the definition of Gl(𝔱, 𝔰, P). That is, if F_{1} and F_{2} are frames over separate points x_{1} and x_{2} in X, the action of S^{1} identifies (F_{1}, F_{2}) with (e^{iθ}F_{1}, e^{iθ}F_{2}). There can be u ∈ G_{𝔰} with u(x_{1}) ≠ u(x_{2}). For such u ∈ G_{𝔰}, (uF_{1}, uF_{2}) would not be identified with (F_{1}, F_{2}) by the S^{1} action.

Proof of Proposition 9.12. By Lemmas 9.12, 9.13, and 9.14, we see that the Euler class of the pullback, by the map (9.25), of the restriction of Υ_{𝔱,𝔰}^{i}/S^{1} to the space (9.24) is given by

(9.40)  ∏_{P ∈P} π_{G̃l,P}^{∗} e(Υ_{𝔱,𝔰}^{i}(P)/S^{1}) = ∏_{P ∈P} π_{G̃l,P}^{∗} e(π_{X,𝔰}^{∗}𝕃_{𝔰}^{∗} ⊗ Υ_{X,P}^{i} /S^{1}).

Now, by the splitting principal, we can assume that Υ_{X,P}^{i} /S^{1} = ⊕_{i∈P} L_{i}, where L_{i} is a complex line bundle and all the symmetric polynomials in c_{1}(L_{i}) are given by polynomials in c(𝔱) and, by (9.36), in ν − 2π_{X,𝔰}^{∗}c_{1}(𝕃_{𝔰}). Observe that π_{X,𝔰} ∘ π_{G̃l,P} = π_{𝔰,i} ∘ π_{X}, so the symmetric polynomials in π_{G̃l,P}^{∗} π_{X,𝔰}^{∗}𝕃_{𝔰} are given by the Chern classes ι̃_{P}^{∗}π_{X}^{∗}c_{𝔰,ℓ,i} defined in (9.21). Hence, we can rewrite (9.40) as

(9.41)  = ∏_{P ∈P} ∏_{i∈P} (ι̃_{P}^{∗}π_{𝔰,i}^{∗}c_{1}(𝕃_{𝔰})) π_{G̃l,P}^{∗} c_{1}(L_{i}).

One then verifies the preceding product can be expressed in terms of symmetric polynomials in c_{1}(L_{i}) and π_{𝔰,i}^{∗}c_{1}(𝕃_{𝔰}). □

9.5. The global instanton obstruction Euler class. We now piece together the local computations of the preceding section to give a global characterization of the instanton obstruction Υ_{𝔱,𝔰}^{i}/S^{1} → M_{𝔱,𝔰}^{vir}/S^{1}.

Proposition 9.16. The Euler class of the instanton obstruction bundle Υ_{𝔱,𝔰}^{i}/S^{1} → M_{𝔱,𝔰}^{vir}/S^{1} is given by a polynomial in ι^{∗}ν, ι^{∗}π_{X}^{∗}S^{ℓ}(c(𝔱)), and ι^{∗}π_{X,𝔰}^{∗}c_{𝔰,ℓ,i} which is otherwise independent of X.

Proof. The construction of the bundle Υ_{spl,κ}^{i} → M_{spl,κ}^{s,♮}(δ) and the Spin^{u}(4) equivariance of the diagram (7.12) implies that the particular classifying maps of the local instanton obstruction bundles which admit the factorization (9.39) can be chosen to be equal on the overlap given in Lemma 8.1. Hence, the local equality of cohomology classes given in Proposition 9.11 is an equality of cocycles and thus an equality of global cohomology classes. □

Definition 9.17. We write e(Υ_{𝔱,𝔰}^{i}/S^{1}) for the real cohomology class given by the Euler class of Υ_{𝔱,𝔰}^{i}/S^{1} as described in Proposition 9.16.

9.6. Extensions. We define extensions of the cohomology classes γ_{M}^{∗}μ_{p}(β), γ_{M}^{∗}μ_{c}, and e(Υ_{𝔱,𝔰}^{i}/S^{1}) from the top stratum M_{𝔱,𝔰}^{vir}/S^{1} to M̄_{𝔱,𝔰}^{vir,∗}/S^{1} as follows. Definition 9.18. For β ∈ H_{•}(X; ℝ), let

(9.42)  μ̄_{p}(β) ∈ H^{4−•}(M̄_{𝔱,𝔰}^{vir,∗}/S^{1}; ℝ),

be the cohomology class obtained by omitting the pullback ι^{∗} on the right-hand-side of (9.19). Similarly, let

(9.43)  μ̄_{c} = −ν ∈ H^{2}(M̄_{𝔱,𝔰}^{vir,∗}/S^{1}; ℝ),

be the cohomology class obtained by omitting the pullback ι^{∗} on the right-hand-side of (9.19). Let

(9.44)  ē_{i} ∈ H^{2ℓ}(M̄_{𝔱,𝔰}^{vir,∗}/S^{1}; ℝ)

be the extensions of e(Υ_{𝔱,𝔰}^{i}/S^{1}) obtained by omitting the pullback ι^{∗} from the expression in Proposition 9.16. Remark 9.19. Note that there is no need to extend the Euler class of the Seiberg-Witten obstruction bundle, e(Υ_{𝔱,𝔰}^{s}/S^{1}), as it is already defined on M̄_{𝔱,𝔰}^{vir,∗}/S^{1} by Lemma 9.10.

10. The link

We now use the Thom-Mather structure defined in §6.6 to construct an ambient link, L̄_{𝔱,𝔰}^{vir}, given by the boundary of a neighborhood of M_{𝔰} × Sym^{ℓ}(X) in M̄_{𝔱,𝔰}^{vir}/S^{1} and then the actual link L̄_{𝔱,𝔰}, given by the boundary of a neighborhood of M_{𝔰} × Sym^{ℓ}(X) in M̄_{𝔱}/S^{1}.

We construct the link in §10.1 using the tubular distance functions t(𝔱, 𝔰, P_{i}) constructed in Lemma 6.15. We use our understanding of the overlap maps to show that the ambient link L̄_{𝔱,𝔰}^{vir} can be decomposed into pieces (enumerated by the strata of Sym^{ℓ}(X)) each of which admits a fiber bundle structure. Each piece is a smoothly-stratified space, with strata given by smooth manifolds with corners. The intersections of the different pieces are given by these corners. In §10.2, we show that the intersection pairing appearing in (2.51) can be computed by a pairing of the appropriate cohomology classes with the fundamental class of the ambient link. We reduce the pairing with the fundamental class of the ambient link to one with a simpler subspace, B̄L_{𝔱,𝔰}^{vir} in §10.3. This subspace is also a union of pieces each of which is a smoothly-stratified space with the strata given by manifolds with corners. In addition, each piece of B̄L_{𝔱,𝔰}^{vir} is a fiber bundle over a stratum of Sym^{ℓ}(X). In §10.4, we describe the intersections of the pieces of B̄L_{𝔱,𝔰}^{vir} which are given by the corners as described above.

For use in this section and the following, pick a representative from each conjugacy class of partitions of N_{ℓ} and enumerate them in such a way that

Σ(X^{ℓ}, P_{i}) ⊂ clΣ(X^{ℓ}, P_{j}) only if i < j.

Write these partitions as P_{1}, P_{2}, …, P_{r} so that Σ(X^{ℓ}, P_{1}) is the lowest stratum and Σ(X^{ℓ}, P_{r}) is the highest.

10.1. Defining the link. We define the link by first constructing an ambient link in §10.1.1. We discuss the orientations of the link in §10.1.2 which leads to a formal statement of the cobordism formula (2.51) in Theorem 10.7. Then in §10.1.3, we give a more detailed description of the pieces of the link, showing how they admit a fiber bundle structure.

10.1.1. The ambient link. The ambient link is defined as the boundary of the neighborhood of M_{𝔰}×Sym^{ℓ}(X) in M̄_{𝔱,𝔰}^{vir}/S^{1}. The subspace M_{𝔰}×Sym^{ℓ}(X) of M̄_{𝔱,𝔰}^{vir}/S^{1} is the intersection of the zero-locus of the function defined by the length of the fiber vector of the vector bundle N(δ) → M_{𝔰},

(10.1)  t_{N} : M̄_{𝔱,𝔰}^{vir}/S^{1} → [0, δ],

with the union of the zero-loci of the tubular distance function t(𝔱, 𝔰, P_{i}) defined on U(𝔱, 𝔰, P_{i}) in Lemma 6.15. We will define the ambient link as the union of two pieces:

(10.2)  L̄_{𝔱,𝔰}^{vir} = L̄_{𝔱,𝔰}^{vir,s} ∪ L̄_{𝔱,𝔰}^{vir,i}.

The first piece, L̄_{𝔱,𝔰}^{vir,s}, is defined to be the codimension-one subspace

(10.3)  L̄_{𝔱,𝔰}^{vir,s} = t_{N}^{−1}(δ) ⊂ M̄_{𝔱,𝔰}^{vir}/S^{1}.

The definition of the instanton component of the link, L̄_{𝔱,𝔰}^{vir,i}, is more involved as it will be the union of the hypersurfaces of the local functions t̃(𝔱, 𝔰, P). Pick a small constant ε_{i} for each stratum Σ(X^{ℓ}, P_{i}) (these will be chosen to be generic with respect to requirements obtained later). Then, we define the local link by

(10.4)  L̄_{𝔱,𝔰}^{vir,i}(P_{j}) = U(𝔱, 𝔰, P_{j}) ∩ t(𝔱, 𝔰, P_{j})^{−1}(ε_{j}) − ∪_{i≠j}t(𝔱, 𝔰, P_{i})^{−1}([0, ε_{i})).

Taking the ‘hypersurface’, t(𝔱, 𝔰, P_{j})^{−1}(ε_{j}) makes the subspace L̄_{𝔱,𝔰}^{vir,i}(P_{j}) a link for M_{𝔰} × Σ(X^{ℓ}, P_{j}). Removing the subspaces t(𝔱, 𝔰, P_{i})^{−1}([0, ε_{i})) from U(𝔱, 𝔰, P_{j}) ensures that L̄_{𝔱,𝔰}^{vir,i}(P_{j}) is disjoint from the other strata of reducibles M_{𝔰} × Σ(X^{ℓ}, P_{i}) (for i ≠ j). We then define the instanton component of the link:

(10.5)  L̄_{𝔱,𝔰}^{vir,i} = ∪_{i} L̄_{𝔱,𝔰}^{vir,i}(P_{i}).

The next lemma follows immediately from the construction. Lemma 10.1. The ambient link L̄_{𝔱,𝔰}^{vir} defined in (10.2), (10.3), and (10.5) is the boundary of a neighborhood of M_{𝔰} × Sym^{ℓ}(X) in M̄_{𝔱,𝔰}^{vir}/S^{1}. For generic choices of the constants ε_{i}, the intersection of L̄_{𝔱,𝔰}^{vir} with each stratum of M̄_{𝔱,𝔰}^{vir}/S^{1} is a smooth manifold with corners.

We can now define the link: Definition 10.2. The link, L̄_{𝔱,𝔰}, of M_{𝔰}×Sym^{ℓ}(X) in M̄_{𝔱}/S^{1} is defined to be the intersection of the ambient link L̄_{𝔱,𝔰}^{vir} defined in (10.2) with the zero-locus of the obstruction section.

We refer to the intersection of any two or more of the pieces L̄_{𝔱,𝔰}^{vir,s} or L̄_{𝔱,𝔰}^{vir,i}(P_{j}) as an edge. For generic choices of the parameters δ, ε_{i}, the intersection of each of these edges with the strata of M̄_{𝔱,𝔰}^{vir}/S^{1} will also be a smooth manifold with corners. Lemma 10.3. For generic values of the parameters δ, ε_{i} used to define the ambient link, the link L̄_{𝔱,𝔰} is the boundary of a neighborhood of M_{𝔰} × Sym^{ℓ}(X) in M̄_{𝔱}/S^{1}. The link L̄_{𝔱,𝔰} is smoothly-stratified subspace of M̄_{𝔱}/S^{1}; the intersection of L̄_{𝔱,𝔰} with each stratum of M̄_{𝔱}/S^{1} and each edge of L̄_{𝔱,𝔰}^{vir} is a smooth, oriented manifold with corners. The intersections of L̄_{𝔱,𝔰} with the lower strata of M̄_{𝔱}/S^{1} have codimension at least two in L̄_{𝔱,𝔰}.

The following result then translates immediately from [11]. Lemma 10.4. Assume w ∈ H^{2}(X; ℤ) is such that w (mod 2) is good in the sense of Definition 2.3. Given a spin^{u} structure 𝔱 on X and a spin^{c} structure 𝔰 on X satisfying ℓ(𝔱, 𝔰) ≥ 0 and w_{2}(𝔱) ≡ w (mod 2), there are positive constants ε_{0} and δ_{0} such that the following hold for all generic choices of ε_{i} ≤ ε_{0} and δ ≤ δ_{0} defining L̄_{𝔱,𝔰}^{vir}.

• The link L̄_{𝔱,𝔰} is disjoint from M̄_{κ}^{w} and M̄_{𝔱}^{red} in the stratification (2.13) of M̄_{𝔱}/S^{1}.

• For all z ∈ 𝔸(X) and positive integers η satisfying (10.10), and for V̄(z) and W̄ the geometric representatives discussed in §2.4, the intersection

(10.6)  V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱,𝔰}

is a finite collection of points contained in the top stratum L_{𝔱,𝔰} of L̄_{𝔱,𝔰} ⊂ M̄_{𝔱}/S^{1}, and is disjoint from the edges of L̄_{𝔱,𝔰}^{vir}.

10.1.2. Orientations of the link. To define an intersection number from the intersection (10.6), it is necessary to discuss orientations of the link. An orientation for M_{𝔱} determines one for L_{𝔱,𝔰} through the convention introduced in [16, Equations (2.16), (2.16) & (2.25)] by considering L_{𝔱,𝔰} as a boundary of (M_{𝔱} − γ(M_{𝔱,𝔰}^{vir}))/S^{1}. Specifically, at a point [A, Φ] ∈ L_{𝔱,𝔰}, if

• r⃗ ∈ T M_{𝔱}^{∗,0} is an outward-pointing radial vector with respect to the open neighborhood M_{𝔱} ∩ γ(M_{𝔱,𝔰}^{vir}) and complementary to the tangent space of L_{𝔱,𝔰},

• v_{S^{1}} ∈ T M_{𝔱}^{∗,0} is tangent to the orbit of [A, Φ] under the (free) circle action (where S^{1} ⊂ ℂ has its usual orientation), and

• λ_{M} ∈ det(T M_{𝔱}^{∗,0}) is an orientation for T M_{𝔱} at [A, Φ], then we define an orientation λ_{L} for T L_{𝔱,𝔰} at [A, Φ] by

(10.7)  λ_{M} = −v_{S^{1}} ∧ r⃗ ∧ λ̃_{L},

where the lift λ̃_{L} ∈ Λ^{max −2}(T M_{𝔱}^{∗,0}) at [A, Φ] of λ_{L} ∈ det(T L_{𝔱,𝔰}) ⊂ Λ^{max −1}(T (M_{𝔱}^{∗,0}/S^{1})), obeys π_{∗}λ̃_{L} = λ_{L}, if π : M_{𝔱} → M_{𝔱}/S^{1} is the quotient map. Definition 10.5. If O is an orientation for M_{𝔱}, we call the orientation for L_{𝔱,𝔰} related to O by equation (10.7) the boundary orientation defined by O.

The standard orientation for L̄_{𝔱,𝔰} is defined in [11, Definition 3.12]. The standard orientation arises from the local topology of L̄_{𝔱,𝔰}^{vir} and thus is more natural for computations on this space. The standard and boundary orientations are related in the following. Lemma 10.6. [11, Lemmas 3.13 & 3.14] If 𝔱 and 𝔱_{ℓ} are spin^{u} structures on X satisfying

p_{1}(𝔱) = p_{1}(𝔱_{ℓ}) − 4ℓ, c_{1}(𝔱) = c_{1}(𝔱_{ℓ}), and w_{2}(𝔱) = w_{2}(𝔱_{ℓ}).

Let Ω be a homology orientation and let w be an integral lift of w_{2}(𝔱). If 𝔱_{ℓ} admits a splitting 𝔱_{ℓ} = 𝔰 ⊕ 𝔰 ⊗ L, then the standard orientation for L̄_{𝔱,𝔰} and the boundary orientation for L̄_{𝔱,𝔰} defined through the orientation O^{asd}(Ω, w) for M_{𝔱} differ by a factor of

(10.8)  (−1)^{o_{𝔱}(w,𝔰)}, where o_{𝔱}(w, 𝔰) = \frac{1}{4}(w − c_{1}(L))^{2}.

For all z ∈ 𝔸(X) and positive integers η satisfying (10.10), and for V̄(z) and W̄ the geometric representatives discussed in §2.4, we can then define the intersection number

(10.9)  # (V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱,𝔰})

to be the oriented count of points in the intersection (10.6), using the standard orientation of L̄_{𝔱,𝔰}. This yields the following formal expression of the cobordism formula. Theorem 10.7. Let 𝔱 be a spin^{u} structure on a smooth, oriented four-manifold X. Let z ∈ 𝔸(X) and η a non-negative integer satisfy

(10.10)  deg(z) + 2η = dim M_{𝔱}^{∗,0} − 2.

Assume there is w ∈ H^{2}(X; ℤ) satisfying w_{2}(𝔱) ≡ w (mod 2) and which is good in the sense of Definition 2.3. Then the intersection numbers (10.9) satisfy

(10.11) # (V̄(z) ∩ W̄^{n_{a}−1} ∩ L̄_{𝔱,κ}^{w}) = − ∑_{𝔰∈Spin^{c}(X)}(−1)^{o_{𝔱}(w,𝔰)} # (V̄(z) ∩ W̄^{n_{a}−1} ∩ L̄_{𝔱,𝔰}) ,

where L̄_{𝔱,κ}^{w} is the link of the anti-self-dual connections in M̄_{𝔱}/S^{1} from [15, Definition 3.7].

10.1.3. The fiber bundle structure. We now describe how the fiber bundle structure of the pieces L̄_{𝔱,𝔰}^{vir,i}(P_{j}) of the link. Observe that the functions t(𝔱, 𝔰, P_{j}), while nominally defined on the subspaces U(𝔱, 𝔰, P) of M̄_{𝔱,𝔰}^{vir}/S^{1} can also be understood as functions on subsets of Sym^{ℓ}(X) by virtue of the inclusion,

N(δ) × Sym^{ℓ}(X) ⊂ M̄_{𝔱,𝔰}^{vir}/S^{1},

and by virtue of the equality in item (2) of Lemma 6.15, the restriction of t(𝔱, 𝔰, P_{j}) to

(N(δ) × Sym^{ℓ}(X)) ∩ U(𝔱, 𝔰, P)

is equal to the pullback by the projection

N(δ) × Sym^{ℓ}(X) → Sym^{ℓ}(X).

of the function t(X^{ℓ}, P) defined in Lemma 4.37.

With this understanding, define

(10.12)  K_{j} = Σ(X^{ℓ}, P_{i}) − ∪_{i<j} t(X^{ℓ}, P_{i})^{−1}([0, ε_{i})).

Recall that the open subspace U(𝔱, 𝔰, P_{j}) is a neighborhood of N_{𝔱,𝔰}(δ)/S^{1} × Σ(X^{ℓ}, P_{j}) in the fiber bundle

(10.13)  \begin{aligned}∏_{P ∈P} M̄_{spl,|P |}^{s,♮}(δ_{P}) & ⟶ & Ñ(δ_{P}) ×_{G_{𝔰}×S^{1}} Gl(𝔱, 𝔰, P) \\ π(𝔱,𝔰,P_{j}) & ↓ \\ & & N(δ)/S^{1} × Σ(X^{ℓ}, P_{j}) \end{aligned}

We will write

(10.14)  M̄(P) = ∏_{P ∈P} M̄_{spl,|P |}^{s,♮}(δ_{P})

for the fiber of (10.13).

The first step in describing the fiber bundle structure of L_{𝔱,𝔰}^{vir,i}(P_{j}) in U(𝔱, 𝔰, P_{j}) is the following description of the base of the bundle. Lemma 10.8. The space K_{j} defined in (10.12) is a compact, codimension-zero submanifold with corners of Σ(X^{ℓ}, P_{j}). Then, the subspace

(10.15)  U(𝔱, 𝔰, P_{j}) − ∪_{i<j} t(𝔱, 𝔰, P_{i})^{−1}([0, ε_{i}))

is given by the intersection of U(𝔱, 𝔰, P_{j}) with the restriction of the fiber bundle (10.13) to N(δ)/S^{1} ×K_{j}. In addition, the subspace (10.15) is disjoint from the subspace N(δ)/S^{1} ×K_{i} for all i < j.

Proof. By (6.52), for i < j the function t(𝔱, 𝔰, P_{i}) is constant on the fibers of the projection map π(𝔱, 𝔰, P_{j}). Hence, restricting π(𝔱, 𝔰, P_{j}) to π(𝔱, 𝔰, P_{j})^{−1}(K_{j}) does not change the fibers of this projection. The equality between t(𝔱, 𝔰, P_{i}) and t(X^{ℓ}, P_{j}) in item (2) of Lemma 6.15 then completes the proof. □

The function t(𝔱, 𝔰, P_{j}) on U(𝔱, 𝔰, P_{j}) is defined by a function,

(10.16)  t_{f}(P_{j}) = ∑_{P ∈P_{j}} |P |λ̃_{|P |}^{2} : M̄(P) = ∏_{P ∈P_{j}} M̄_{spl,|P |}^{s,♮}(δ_{P}) → [0, ∞)

(see (6.50)) on the fibers of (10.13). Because t_{f}(P_{j}) is G(P_{j})-invariant it extends to a function on (10.13); this function is t(𝔱, 𝔰, P_{j}). Because the open subspace U(𝔱, 𝔰, P_{j}) is defined by a subspace of the fiber bundle Gl(𝔱, 𝔰, P_{j}) in (10.13), the following lemma is necessary to ensure that working with the subspace U(𝔱, 𝔰, P_{j}) does not restrict the fibers of the local link L̄_{𝔱,𝔰}^{vir,i}(P_{j}).

Lemma 10.9. For any compact K ⋐ Σ(X^{ℓ}, P_{j}), there is ε_{K} such that the intersection

U(𝔱, 𝔰, P_{i}) ∩ π(𝔱, 𝔰, P_{j})^{−1} (N(δ)/S^{1} × K) ,

contains t(𝔱, 𝔰, P_{j})^{−1}([0, ε_{K})) ∩ π(𝔱, 𝔰, P_{j})^{−1} (N(δ)/S^{1} × K).

Proof. The lemma follows immediately from the compactness of K. □

Lemma 10.9 allows us to define a link of N(δ)/S^{1}×K_{j} using a subspace of (10.13) defined by replacing the fiber M̄(P_{j}) with the subspace

(10.17)  M̄(P_{j}, ε_{j}) = t_{f}(P_{j})^{−1}(ε_{j}) ⊂ M̄(P_{j})

where t_{f}(P_{j}) is defined in (10.16). The resulting subspace will be t(𝔱, 𝔰, P_{j})^{−1}(ε_{j}): Lemma 10.10. Let ε_{j} > 0 be a small constant given by Lemma 10.9 for the space K_{j} defined in (10.12). The subspace

(10.18)  t(𝔱, 𝔰, P_{j})^{−1}(ε_{j}) − ∪_{i<j} t(𝔱, 𝔰, P_{i})^{−1}([0, ε_{i}))

is the codimension-one subspace of U(𝔱, 𝔰, P_{j}), disjoint from N(δ)/S^{1} × Σ(X^{ℓ}, P_{i}) for all i ≤ j and is given by

(10.19)  \begin{aligned}M̄(P_{j}, ε_{j}) & ⟶ & Ñ(δ) ×_{G_{𝔰}×S^{1}} Fr(𝔱, 𝔰, P_{j}, g_{P_{j}})|_{K_{j}} ×_{G(P_{j})} M̄(P, ε_{j}) \\ π(𝔱,𝔰,P_{j}) & ↓ \\ & & N(δ)/S^{1} × K_{j} \end{aligned}

where M̄(P_{j}, ε_{j}) is defined in (10.17).

Proof. The subspace (10.18) is disjoint from the subspaces N(δ)/S^{1} × Σ(X^{ℓ}, P_{i}) for i < j by Lemma 10.8 and from N(δ)/S^{1} × Σ(X^{ℓ}, P_{j}) because

N(δ)/S^{1} × Σ(X^{ℓ}, P_{j}) = t(𝔱, 𝔰, P_{j})^{−1}(0).

The subspace (10.18) is a sub-fiber bundle of (10.13) because, as noted above, the function t(𝔱, 𝔰, P_{j}) is defined by extending a function on the fibers over the bundle. □

The subspace (10.18) still intersects M_{𝔰} × Sym^{ℓ}(X), but only along higher strata M_{𝔰} × Σ(X^{ℓ}, P_{i}) where i > j. We remove this intersection by removing neighborhoods of the subspace t(𝔱, 𝔰, P_{i})^{−1}(0) ∩ U(𝔱, 𝔰, P_{j}), giving the definition of L̄_{𝔱,𝔰}^{vir,i}(P_{j}) found in (10.4). To understand the subspace L̄_{𝔱,𝔰}^{vir,i}(P_{j}) of (10.18), we must examine the function t(𝔱, 𝔰, P_{i}) on U(𝔱, 𝔰, P_{j}) ∩ U(𝔱, 𝔰, P_{i}).

Recall from Proposition 6.9 that the intersection U(𝔱, 𝔰, P) ∩ U(𝔱, 𝔰, P′) (where P < P′) is given by an open subspace of the fiber bundle

(10.20)  Ñ(δ) ×_{G_{𝔰}×S^{1}} Fr(𝔱, 𝔰, P, g_{P}) ×_{G(P)} ∐_{P′′∈[P<P′]} M̄(P, P′′) ; ↓ ; N(δ)/S^{1} × Σ(X^{ℓ}, P)

where the fiber has components (up to a quotient by the symmetric subgroup 𝔖(P))

(10.21)  M̄(P, P′′) = ∏_{P ∈P} (Δ^{∘}(Z_{|P |}(δ_{P}), P′′_{P}) × ∏_{Q∈P′′_{P}} M̄_{spl,|Q|}^{s,♮}(δ_{Q})) .

There is an 𝔖(P)-equivariant projection map,

(10.22)  ∐_{P′′∈[P<P′]} M̄(P, P′′) → ∐_{P′′∈[P<P′]} ∏_{Q∈P′′} M̄_{spl,|Q|}^{s,♮}(δ_{Q}).

If λ̃_{Q} is the modified scale parameter on M̄_{spl,|Q|}^{s,♮}(δ_{Q}) defined in (5.69)), then the function,

(10.23)  ∑_{Q∈P′} |Q|λ̃_{Q}^{2} : ∏_{Q∈P′′} M̄_{spl,|Q|}^{s,♮}(δ_{Q}) → [0, ∞)

extends to a unique 𝔖(P)-invariant function on the image of the projection (10.22). The pullback of this 𝔖(P)-invariant function by the projection map (10.22) defines a function,

(10.24)  t_{o,f}(P, [P′]) : ∐_{P′′∈[P<P′]} M̄(P, P′′) → [0, ∞).

The function t_{o,f}(P, [P′]) is invariant under the action of the group G(P) and thus defines a function on the bundle (10.20):

(10.25) t̃_{o}(P, [P′]) : Ñ(δ) ×_{G_{𝔰}×S^{1}} Fr(𝔱, 𝔰, P, g_{P}) ×_{G(P)} ∐_{P′′∈[P<P′]} M̄(P, P′′) → [0, ∞).

Lemma 10.11. Let P < P′ be partitions of N_{ℓ}. Let ρ_{P,[P′]}^{N,𝔱,𝔰,u} be the upwards overlap map

appearing in (6.40). The pullback t(𝔱, 𝔰, P′) ∘ ρ_{P,[P′]}^{N,𝔱,𝔰,u} is defined on the subspace U(𝔱, 𝔰, P) ∩ U(𝔱, 𝔰, P′) of the fiber bundle (10.20) and is equal to the function t̃_{o}(P, [P′]) from (10.25).

Proof. The lemma follows immediately from the definition of the function t(𝔱, 𝔰, P′) and the observation that the overlap map ρ_{P,[P′]}^{N,𝔱,𝔰,u} is the identity on the spaces M̄_{spl,|Q|}^{s,♮}(δ_{Q}). □

To understand the space (10.4), we now discuss the behaviour of the function t̃_{o}(P, [P′]) under the downwards transition map ρ_{P,[P′]}^{𝔱,𝔰,d} from (6.40). On the component of the inter-

section U(𝔱, 𝔰, P) ∩ U(𝔱, 𝔰, P′) corresponding to the partition P′′ ∈ [P < P′], the map ρ_{P,[P′]}^{𝔱,𝔰,d} is defined by the inclusion of fibers (see the discussion around (6.35)) defined the product (over P ∈ P) of splicing maps (of connections to the trivial connection on ℝ^{4}):

(10.26)  ∐_{P′′∈[P<P′]} M̄(P, P′′) → M̄(P) = ∏_{P ∈P} M̄_{spl,|P |}^{s,♮}(δ_{P}).

Let

(10.27)  U_{f}(P, [P′]) ⊂ M̄(P)

be the image of the inclusion (10.26). (Note that the map (10.26) is actually injective only after taking the quotient of the domain and image by 𝔖(P); we omit that quotient from the notation since it is not relevant to this discussion.) Let

(10.28)  t_{f}(P, [P′]) : U_{f}(P, [P′]) → [0, ∞)

be the function whose pullback by the inclusion (10.26) is equal to the function t_{o,f}(P, [P′]) from (10.24). We can characterize the function t_{f}(P, [P′]) as follows. Let

t(Θ, P′_{P}) : U(Θ, P′_{P}) ⊂ M̄_{spl,|P |}^{s,♮} → [0, ∞),

be the tubular distance function defined in (5.73). These function (pulled back by the appropriate projections) define a function

(10.29)  ∑_{P ∈P} |P |t(Θ, P′_{P})^{2} : M̄(P) = ∏_{P ∈P} M̄_{spl,|P |}^{s,♮} → [0, ∞),

which is independent of P′ ∈ [P < P′] and we have the following: Lemma 10.12. The function t_{f}(P, [P′]) defined in (10.28) is equal to the function defined in (10.29).

Proof. The lemma follows form the definition of t_{f}(P, [P′]) and the definition (5.73). □

We then have: Lemma 10.13. For i > j, the function t(𝔱, 𝔰, P_{i}) on the subspace U(𝔱, 𝔰, P_{i}) ∩ U(𝔱, 𝔰, P_{j}) is defined by the extension of the G(P)-invariant function t_{f}(P, [P′]) from the fiber U_{f}(P, [P′]) to the subspace U(𝔱, 𝔰, P_{i}) ∩ U(𝔱, 𝔰, P_{j}) of the fiber bundle (10.13).

For all k > j we define

(10.30)  M̄(P_{j}, [P_{k}], ε) = M̄(P_{j}) − t_{f}(P_{j}, [P_{k}])^{−1}([0, ε)).

Let ε_{j} = (ε_{j}, ε_{j+1}, …, ε_{r}). Then define

(10.31)  M̄(P_{j}, ε_{j}) = M̄(P_{j}, ε_{j}) ∩ (∩_{k>j}M̄(P_{j}, [P_{k}], ε_{k})) ⊂ M̄(P).

We can then conclude with the following description of the fiber bundle structure of L̄_{𝔱,𝔰}^{vir,i}(P_{j}).

Lemma 10.14. The local instanton link L̄_{𝔱,𝔰}^{vir,i}(P_{j}) defined in (10.4) is the subbundle of the restriction of the fiber bundle (10.13) to N(δ)/S^{1} × K_{j} defined by the subspace (10.31) of the fiber M̄(P_{j}):

(10.32)  \begin{aligned}M̄(P_{j}, ε_{j}) & ⟶ & Ñ(δ_{P}) ×_{G_{𝔰}×S^{1}} Fr(𝔱, 𝔰, P_{j}, g_{P_{j}}) ×_{G(P_{j})} M̄(P, ε) \\ π(𝔱,𝔰,P_{j}) & ↓ \\ & & N(δ)/S^{1} × K_{j} \end{aligned}

This local instanton link is disjoint from N(δ)/S^{1} × Sym^{ℓ}(X).

10.2. Duality and the link. We now discuss how the intersection number,

(10.33)  # (V̄(z) ∩ W̄^{δ_{c}} ∩ L̄_{𝔱,𝔰})

can be expressed in terms of pairing a cohomology class with a homology class. The proof of the following is analogous to that [11, Proposition 5.2] Theorem 10.15. For β ∈ H_{•}(X; ℝ), let

μ̄_{p}(β), μ̄_{c}, and ē_{i} ∈ H^{•}(M̄_{𝔱,𝔰}^{vir,∗}/S^{1}; ℝ)

be the cohomology classes from Definition 9.18. Let ē_{s} = e(Υ_{𝔱,𝔰}^{s}/S^{1}) be the Euler class of the Seiberg-Witten obstruction bundle. If d(𝔱) = dim M_{𝔱,𝔰}^{∗,0}, then let [L_{𝔱,𝔰}^{vir}] ∈ H_{d(𝔱)−2}(M̄_{𝔱,𝔰}^{vir,∗}/S^{1}; ℝ) be the homology class defined in (10.35). Then

(10.34)  # (V̄(z) ∩ W̄^{δ_{c}} ∩ L̄_{𝔱,𝔰}) = ⟨μ̄_{p}(z) ⌣ μ̄_{c}^{δ_{c}} ⌣ ē_{i} ⌣ ē_{s}, [L_{𝔱,𝔰}^{vir}]⟩.

10.2.1. The homology class of the ambient link. We now define the homology class of the ambient link,

[L_{𝔱,𝔰}^{vir}] ∈ H_{d(𝔱)−2}(M̄_{𝔱,𝔰}^{vir,∗}/S^{1}; ℝ).

The union of the lower strata in L_{𝔱,𝔰}^{vir} will be denoted L_{𝔱,𝔰}^{sing} ⊂ L_{𝔱,𝔰}^{vir}. By [24, Propositions 3.2], L_{𝔱,𝔰}^{sing} ⊂ L_{𝔱,𝔰}^{vir} is a neighborhood deformation retract, so there is an open neighborhood U ⊂ L_{𝔱,𝔰}^{vir} such that

(1) L_{𝔱,𝔰}^{vir} − U is a topological manifold with boundary,

(2) The neighborhood U retracts onto L_{𝔱,𝔰}^{sing}.

The neighborhood U and the retraction of U onto L_{𝔱,𝔰}^{sing} can also be constructed by noting that (i) the property of being a neighborhood deformation retraction is a local property and that (ii) the pre-image of the lower strata of M̄_{κ}^{s,♮}(δ) in the bubbletree compactification of M_{κ}^{s,♮}(δ) as defined in [4] form the boundary of a topological manifold and thus are a neighborhood deformation retraction. Let

ȷ : (L_{𝔱,𝔰}^{vir}, ∅) → (L_{𝔱,𝔰}^{vir}, U)

be the map of pairs. Because U retracts to the codimension-four subspace L_{𝔱,𝔰}^{sing},

ȷ_{∗} : H_{d(𝔱)−2} (L_{𝔱,𝔰}^{vir}, ∅) → H_{d(𝔱)−2} (L_{𝔱,𝔰}^{vir}, U)

is an isomorphism. If

[L_{𝔱,𝔰}^{vir} − U, ∂(L_{𝔱,𝔰}^{vir} − U)] ∈ H_{d(𝔱)−2} (L_{𝔱,𝔰}^{vir}, U)

is the relative fundamental class of the(topological) manifold with boundary L_{𝔱,𝔰}^{vir} − U, then we define

(10.35)  [L_{𝔱,𝔰}^{vir}] = ȷ_{∗}^{−1}[L_{𝔱,𝔰}^{vir} − U, ∂(L_{𝔱,𝔰}^{vir} − U)],

to be the fundamental class of the ambient link.

Proof of Theorem 10.15. The proof is analogous to that of [11, Proposition 5.2] using relative Euler classes of the obstruction sections and representatives of the cohomology with compact support along the geometric representatives. □

10.3. Reducing to a submanifold. We now show how to reduce pairings with the homology class [L_{𝔱,𝔰}^{vir}] defined in (10.35) to pairings with the homology class of a subspace which eliminates the topology of the normal bundle N(δ) → M_{𝔰}.

The deformation retraction N(δ) → M_{𝔰}. defines a deformation retraction of L̄_{𝔱,𝔰}^{vir,i} to the subspace

(10.36)  B̄L_{𝔱,𝔰}^{vir} = ∪_{i} B̄L_{𝔱,𝔰}^{vir}(P_{i}) = ∪_{i} (t_{N}^{−1}(0) ∩ L_{𝔱,𝔰}^{vir,i}(P_{i}))

where the function t_{N} is defined in (10.1). Essentially, B̄L_{𝔱,𝔰}^{vir}(P_{i}) is defined by replacing Ñ(δ) with M̃_{𝔰} in the definition (10.4). Hence, we have the following fiber bundle description of B̄L_{𝔱,𝔰}^{vir}(P_{i}). Lemma 10.16. The space

(10.37)  B̄L_{𝔱,𝔰}^{vir}(P_{j}) = t_{N}^{−1}(0) ∩ L̄_{𝔱,𝔰}^{vir,i}(P_{j})

is the restriction of the fiber bundle (10.32) to the subspace

M_{𝔰} × K_{j} ⊂ N(δ)/S^{1} × K_{j}

of the base and is thus given by:

(10.38)  \begin{aligned}M̄(P_{j}, ε_{j}) & ⟶ & M̃_{𝔰} ×_{G_{𝔰}×S^{1}} Fr(𝔱, 𝔰, P_{j}, g_{P_{j}}) ×_{G(P_{j})} M̄(P, ε) \\ π(𝔱,𝔰,P_{j}) & ↓ \\ & & M_{𝔰} × K_{j} \end{aligned}

Let B̄L_{𝔱,𝔰}^{vir} ⊂ L̄_{𝔱,𝔰}^{vir} be the subspace defined in (10.36). We observe that L̄_{𝔱,𝔰}^{vir,i} is a disk bundle over B̄L_{𝔱,𝔰}^{vir}. A fundamental class,

(10.39)  [B̄L_{𝔱,𝔰}^{vir}] ∈ H_{d(𝔱)−2r_{N} −2}(L̄_{𝔱,𝔰}^{vir}; ℝ),

(where r_{N} is the complex rank of the bundle N_{𝔱_{0},𝔰} → M_{𝔰}) is defined exactly as is done in (10.35). We then have: Proposition 10.17. Let d_{s} = dim M_{𝔰}. Let s_{j}(N) ∈ H^{2j}(M_{𝔰}) and r_{N} be the Segre classes and complex rank (respectively) of the bundle N_{𝔱_{0},𝔰} → M_{𝔰}. Let k and m be non-negative integers satisfying k + 2m = d(𝔱) − 2. For any α ∈ H^{k}(M_{𝔰} × Sym^{ℓ}(X); ℝ), we have the equality

(10.40)

⟨ν^{m} ⌣ π_{X,𝔰}^{∗}α, [L̄_{𝔱,𝔰}^{vir}]⟩

= ∑_{j=0}^{d_{2}/2}(−1)^{r_{N} +j}⟨ν^{m−r_{N} −j} ⌣ π_{𝔰}^{∗}s_{j}(N) ⌣ π_{X,𝔰}^{∗}α, [B̄L_{𝔱,𝔰}^{vir}]⟩

Proof. See [11, §5.2]. □

Remark 10.18. The Segre classes s_{i}(N) have been computed under some assumptions on H^{1}(X; ℝ) in [15, Lemma 4.11]. From [15, Theorem 3.29] one can see that in general, these Segre classes will be given by a universal polynomial in μ_{𝔰}(x) and μ_{𝔰}(γ_{i}) with coefficients depending only on the indices n′_{s} and n′′_{s} which depend only on p_{1}(𝔱_{0}) and c_{1}(𝔱_{0}).

10.4. Boundary of link components. Each of the pieces B̄L_{𝔱,𝔰}^{vir}(P_{i}) as P_{i} is a stratified space where each stratum is a smooth manifold with corners. We describe this corner structure by denoting

(10.41)  ∂_{k_{1}} … ∂_{k_{r}} B̄L_{𝔱,𝔰}^{vir}(P_{j}) = (∩_{u=1}^{r}t(𝔱, 𝔰, P_{k_{u}})^{−1}(ε_{k_{u}})) ∩ B̄L_{𝔱,𝔰}^{vir}(P_{j}).

Observe that the intersection of the pieces of B̄L_{𝔱,𝔰}^{vir} is described by the equality:

(10.42)  ∂_{k}B̄L_{𝔱,𝔰}^{vir}(P_{j}) = B̄L_{𝔱,𝔰}^{vir}(P_{j}) ∩ B̄L_{𝔱,𝔰}^{vir}(P_{k}) = ∂_{j}B̄L_{𝔱,𝔰}^{vir}(P_{k})

which follows immediately from the definition. If

clΣ(X^{ℓ}, P_{j}) ∩ Σ(X^{ℓ}, P_{k}) = ∅ and Σ(X^{ℓ}, P_{j}) ∩ clΣ(X^{ℓ}, P_{k}) = ∅,

then the boundaries in (10.42) are empty. By the symmetry of the definition (10.41) with respect to k_{u} and the equality (10.42), it suffices to describe the boundary in (10.41) under the assumption that k_{1} < k_{2} < ⋯ < k_{r} < j. Lemma 10.19. Let K_{j} be the subspace defined in (10.12). For k_{1} < k_{2} < ⋯ < k_{r} < j, define ∂_{k_{1}} … ∂_{k_{r}}K_{j} by

(10.43)  ∂_{k_{1}} … ∂_{k_{r}}K_{j} = (∩_{u=1}^{r}t(X^{ℓ}, P_{k_{u}})^{−1}(ε_{k_{u}})) ∩ K_{j} ⊂ Σ(X^{ℓ}, P_{j}).

Then, the corner ∂_{k_{1}} … ∂_{k_{r}} B̄L_{𝔱,𝔰}^{vir}(P_{j}) is the restriction of the fiber bundle

B̄L_{𝔱,𝔰}^{vir}(P_{j}) → K_{j}

to the subspace ∂_{k_{1}} … ∂_{k_{r}}K_{j} ⊂ K_{j}.

Proof. The lemma follows immediately from the definitions (10.41) and (10.43) and the relation between the functions t(𝔱, 𝔰, P_{i}) and t(X^{ℓ}, P_{i}) given in the proof of Lemma 10.8. □

We further extend the description of the corner (10.41) in Lemma 10.19 by describing the corner ∂_{k_{1}} … ∂_{k_{r}}K_{j} ⊂ K_{j}. We begin by describing the corner ∂_{k}K_{j} where k < j. First, observe that for ε_{k} sufficiently small ∂_{k}K_{j} is in the image of the exponential map with domain an open subspace of

(10.44)  Fr(T X^{ℓ}, P_{k}, g_{P_{k}})|_{K_{k}} ×_{G(T,P)} Z(P_{k})/𝔖(P_{k}),

where the fiber Z(P_{k}) is given by

Z(P_{k}) = ∏_{P ∈P_{k}} Z_{P} (δ_{P}).

The function t(X^{ℓ}, P_{k}) is defined by the G(T, P)-invariant functions on the fiber,

(10.45)  m_{P,[P]} = ∑_{P ∈P} |P |t(Z_{P})^{2} : Z(P_{k}) → [0, ∞)

where t(Z_{P}) is defined in Lemma 4.3. (The motivation for the appearance of two copies of P in m_{P,[P]} will be apparent shortly.) For other partitions P′ with P_{k} < P′, a neighborhood of K_{k} in Σ(X^{ℓ}, P′) is given by restricting the exponential map e(X^{ℓ}, g_{P_{k}}) to the subspace of (10.44) given by the subspace of the fiber,

(10.46)  ∐_{P′′∈[P_{k}<P′]} ∏_{P ∈P_{k}} Δ^{∘}(Z_{P} , P′′_{P}) ⊂ Z(P_{k}).

A neighborhood of this subspace of the fiber Z(P_{k}) is given by the union, over P′′ ∈ [P_{k} < P′], of the images of the exponential map

(10.47)  ∏_{P ∈P_{k}} e(Z_{P} , P′′_{P}) : ∏_{P ∈P_{k}} (Δ^{∘}(Z_{P} , P′′_{P}) × ∏_{P′′∈P′′} Z_{P′′}) → Z(P_{k})

where e(Z_{P} , P′′_{P}) is defined in (4.9). On the domain of the map (10.47), we can define a function

(10.48)  m_{P,[P′]} = ∑_{P′′∈P′′} |P′′|(t(Z_{P′′}) ∘ π_{P′′})^{2}

where π_{P′′} is the projection from the domain of the map (10.47) to Z_{P′′}. We also write m_{P,[P′]} for the pushforward of this function to the image of the map (10.47). One can see that the function m_{P,[P′]} is invariant under the choice of P′′ ∈ [P < P′].

Lemma 10.20. For P ≤ P′, the pullback of the function t(X^{ℓ}, P′) by the exponential map e(X^{ℓ}, P) to the domain (10.44) is equal to the function defined by the G(P)-invariant function m_{P,[P′]} on the fiber Z(P).

Proof. The lemma follows from an examination of the diagram (4.22) and the definition of t(X^{ℓ}, P′). Compare the discussion prior to Lemma 10.11. □

Hence, for ε = (ε_{1}, …, ε_{r}) the choice of parameters used to define the link, we define a subspace of the fiber Z(P_{k}) by

(10.49)  Z(P_{k}, ε) = Z(P_{k}) − ∪_{i≥k}m_{P_{k},[P_{i}]}^{−1}([0, ε_{i})).

For k ≤ k_{1} < ⋯ < k_{r}, we define

(10.50)  ∂_{k_{1}} … ∂_{k_{r}}Z(P_{k}, ε) = Z(P_{k}) ∩ (∩_{i=1}^{r}m_{P_{k},[P_{k_{i}}]}^{−1}(ε_{k_{i}})) .

Finally, recall from Lemma 4.5 that a neighborhood of K_{k} in Σ(X^{ℓ}, P_{j}) is parameterized by the subspace of (10.44) given (up to symmetric group quotients) by the subspace of the fiber:

(10.51)  Z(P_{k}, [P_{j}]) = ∐_{P′′∈[P_{k},P_{j}]} ∏_{P ∈P_{k}} Δ^{∘}(Z_{P} (δ_{P}), P′′_{P}).

Hence, for k ≤ k_{1} < ⋯ < k_{r} < j we define

(10.52)  \begin{aligned}Z(P_{k}, [P_{j}], ε) & = Z(P_{k}, ε) ∩ Z(P_{k}, [P_{j}]), \\ ∂_{k_{1}} … ∂_{k_{r}}Z(P_{k}, [P_{j}], ε) & = ∂_{k_{1}} … ∂_{k_{r}}Z(P_{k}, ε) ∩ Z(P_{k}, [P_{j}]). \end{aligned}

The above discussion yields the following result. Lemma 10.21. For k_{1} < k_{1} < ⋯ < k_{r} < j, the subspace ∂_{k_{1}} … ∂_{k_{r}}K_{j} is given by the image under the exponential map e(X^{ℓ}, P_{k_{1}}) of the subspace

(10.53)  Fr(T X^{ℓ}, P_{k_{1}}, g_{P_{k_{1}}} ) ×_{G(T,P_{k_{1}})} ∂_{k_{1}} … ∂_{k_{r}}Z(P_{k_{1}}, [P_{j}], ε)/𝔖(P_{k_{1}})

of the domain (10.44).

The following relation between the boundaries ∂_{j}K_{i} and ∂_{j}B̄L_{𝔱,𝔰}^{vir}(P_{i}) when i < j will also be useful. Lemma 10.22. For i < j, let M̄(P_{i}, ε_{i}) be the space defined in (10.31). Let t_{f}(P_{i}, [P_{j}]) be the function defined in (10.28). Define

(10.54)  ∂_{j}M̄(P_{i}, ε_{i}) = t_{f}(P_{i}, [P_{j}])^{−1}(ε_{j}) ∩ M̄(P_{i}, ε_{i}) ⊂ M̄(P_{i}, ε_{i}).

Then, the boundary ∂_{j}B̄L_{𝔱,𝔰}^{vir}(P_{i}), defined in (10.41), is the fiber subbundle,

(10.55)  \begin{aligned}∂_{j}M̄(P_{i}, ε_{i}) & ⟶ & M̃_{𝔰} ×_{G_{𝔰}×S^{1}} Fr(𝔱, 𝔰, P_{i}, g_{P_{i}}) ×_{G(P_{i})} ∂_{j}M̄(P_{i}, ε_{i}) \\ π(𝔱,𝔰,P_{i}) & ↓ \\ & & M_{𝔰} × K_{i} \end{aligned}

of the fiber bundle B̄L_{𝔱,𝔰}^{vir}(P_{i}) given in (10.38).

Proof. The lemma follows from the description in Lemma 10.13 of the relation between the functions t(𝔱, 𝔰, P_{j}) and t_{f}(P_{i}, [P_{j}]). □

We now give another description of the fiber ∂_{j}M̄(P_{i}, ε_{i}) appearing in (10.54). First, because the function t_{f}(P_{i}, [P_{j}]) is defined on U_{f}(P_{i}, [P_{j}]) in (10.28), we have the inclusion

∂_{j}M̄(P_{i}, ε_{i}) ⊂ U_{f}(P_{i}, [P_{j}]),

where the subspace U_{f}(P_{i}, [P_{j}]) is defined in (10.27). Because U_{f}(P_{i}, [P_{j}]) is defined as the image of the injective (up to symmetric group action) splicing map in (10.26), there is a homeomorphism (up to symmetric group action) between U_{f}(P_{i}, [P_{j}]) and

∐_{P′′∈[P_{i}<P_{j}]} ∏_{P ∈P_{i}} (Δ^{∘}(Z_{|P |}(δ_{P}), P′′_{P}) × ∏_{Q∈P′′_{P}} M̄_{spl,|Q|}^{s,♮}(δ_{Q})) .

This homeomorphism defines a projection map:

(10.56)  U_{f}(P_{i}, [P_{j}]) → Z(P_{i}, [P_{j}])/𝔖(P_{i})

where Z(P_{i}, [P_{j}]) is defined in (10.51). In addition, because

∐_{P′′∈[P_{i}<P_{j}]} ∏_{P ∈P} ∏_{Q∈P′′_{P}} M̄_{spl,|Q|}^{s,♮}(δ_{Q}) = ∐_{P′′∈[P_{i}<P_{j}]} ∏_{Q∈P′′} M̄_{spl,|Q|}^{s,♮}(δ_{Q})

= ∐_{P′′∈[P_{i}<P_{j}]} M̄(P′′),

where M̄(P′′) is defined in (10.14), there is a projection map

(10.57)  U_{f}(P_{i}, [P_{j}]) → ∐_{P′′∈[P_{i}<P_{j}]} M̄(P′′)/𝔖(P_{i}).

We then have: Lemma 10.23. For i < j, if

U_{f}(P_{i}, [P_{j}], ε_{i}) = U_{f}(P_{i}, [P_{j}]) ∩ M̄(P_{i}, ε_{i}).

then the restriction of the projection map (10.56) to U_{f}(P_{i}, [P_{j}], ε_{i}) defines a projection map

(10.58)  U_{f}(P_{i}, [P_{j}], ε_{i}) → Z(P_{i}, [P_{j}], ε)/𝔖(P_{i}),

where the subspace Z(P_{i}, [P_{j}], ε) is defined in (10.52).

Proof. The subspace M̄(P_{i}, ε_{i}) is defined by the functions t_{f}(P_{i}, [P_{k}]) in (10.30) and (10.31). The subspace Z(P_{i}, [P_{j}], ε_{i}) is defined in (10.49) by the functions m_{P_{i},[P_{k}]} in (10.48). The projection map (10.56), is given by a product of the projection maps π(Θ, P′′_{P}) defined in Lemma 5.66. The tubular distance functions t(Θ, P′′_{P}) are constant on the fibers of these projections by (5.67). The lemma then follows from the relation between the functions t_{f}(P_{i}, [P_{k}]) and t(Θ, P′′_{P}) in Lemma 10.12 and the relation between t(Θ, P′_{P}) and the function m_{P,[P′′]} given by the definitions of these function. □

A similar argument yields the following characterization of the fiber ∂_{j}M̄(P_{i}, ε_{i}). Lemma 10.24. For i < j, the restriction of the projection map (10.56) to the fiber ∂_{j}M̄(P_{i}, ε_{i}) appearing in (10.54) defines a projection map

∂_{j}M̄(P_{i}, ε) ⊂ ∂_{j}Z(P_{i}, [P_{j}], ε),

to the subspace ∂_{j}Z(P_{i}, [P_{j}], ε_{i}) defined in (10.52).

We now characterize the fiber of the projection map (10.56) appearing in Lemmas 10.23 and 10.24.

Lemma 10.25. For i < j, the fiber ∂_{j}M̄(P_{i}, ε_{i}) appearing in (10.54) is homeomorphic to the subspace of

(10.59)  ∐_{P′′∈[P_{i}<P_{j}]} ∏_{P ∈P_{i}} (∂_{j}Z(P_{i}, [P_{j}], ε) × ∏_{Q∈P′′_{P}} M̄_{spl,|Q|}^{s,♮}(δ_{Q}))

given by the pre-image of the subspace

∐_{P′′∈[P_{i}<P_{j}]} M̄(P′′, ε_{j})/𝔖(P_{i})

under the projection map (10.57).

Proof. The fiber ∂_{j}M̄(P_{i}, ε_{i}) is defined by the functions t_{f}(P_{i}, [P_{k}]) where k > i. For j > k > i, the constraint t_{f}(P_{i}, [P_{k}])^{−1}((ε_{k}, ∞)), is given, as described in Lemma 10.23, by the restriction of the projection map (10.56) to the subspace ∂_{j}Z(P_{i}, [P_{j}], ε). For k ≥ j > i, the function t_{f}(P_{i}, [P_{k}]) on U_{f}(P_{i}, [P_{j}]) and the function t_{f}(P′′, [P_{k}]) (for P′′ ∈ [P_{i} < P_{j}]) on U_{f}(P′′, [P_{k}]) are both defined by pullback of the function t(𝔱, 𝔰, P_{k}) (see Lemma 10.11). Hence, the constraints t_{f}(P_{i}, [P_{k}])^{−1}((ε_{k}, ∞)) for k ≥ j > i are as described in the statement of the Lemma. □

11. The computation

In this section, we perform the computation leading to a proof of the following theorems which form the technical heart of this paper. Theorem 11.1. Let 𝔱 be a spin^{u} structure on a smooth four-manifold X with b^{1}(X) = 0. Assume that 𝔰 is a spin^{c} structure with M_{𝔰} × Sym^{ℓ}(X) a subset of the space of ideal monopoles IM_{𝔱} defined in (2.12). Let L̄_{𝔱,𝔰} be the link of the reducible SO(3) monopoles given by 𝔰 ∈ Spin^{c}(X) as defined in Definition 10.2. For z = h^{δ−2m}x^{m} ∈ 𝔸(X) and η a non-negative integer satisfying

deg(z) + 2η = dim M_{𝔱} − 2,

let V̄(z) and W̄^{η} be the geometric representatives defined in §2.4. Then

(11.1)  \begin{matrix}# (V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱,𝔰}) \\ = SW_{X}(𝔰) ∑_{i=0}^{min(ℓ,[\frac{δ−2m}{2}])} (q_{δ,ℓ,m,i}(c_{1}(𝔰) − c_{1}(𝔱), c_{1}(𝔱))Q_{X}^{i}) (h). \end{matrix}

where q_{δ,ℓ,m,i} are degree δ − 2m − 2i homogeneous polynomials which are universal functions of the constants given in Theorem 1.1.

We note that a similar result can also be achieved by the methods of this paper without the assumption that b^{1}(X) = 0 and z = h^{δ−2m}x^{m}, but the resulting expression becomes considerably more complicated. While the precise expression for the intersection number in (11.1) is still unknown, we note the following important result, referred to as the ‘multiplicity conjecture’ in [16, 6, 11], which holds even without the simplifying assumptions. Theorem 11.2. Let X be a smooth, oriented four-manifold. Assume that 𝔱 is a spin^{u} structure on X and 𝔰 is a spin^{c} structure on X such that M_{𝔰} × Sym^{ℓ}(X) is contained in the space of ideal monopoles IM_{𝔱} defined in (2.12). Let L̄_{𝔱,𝔰} be the link of M_{𝔰} × Sym^{ℓ}(X) given in Definition 10.2. Then, if SW_{X,𝔰}(ω) vanishes for all ω ∈ 𝔸_{2}(X), then

# (V̄(z) ∩ W̄^{δ} ∩ L̄_{𝔱,𝔰}) = 0.

To prove Theorems 11.1 and 11.2, we begin by observing that Theorem 10.15 gives the equality

(11.2)  # (V̄(z) ∩ W̄^{δ_{c}} ∩ L̄_{𝔱,𝔰}) = ⟨μ̄_{p}(z) ⌣ μ̄_{c}^{δ_{c}} ⌣ ē_{i} ⌣ ē_{s}, [L̄_{𝔱,𝔰}^{vir}]⟩.

The expression for the μ-classes in Corollary 9.9, the expression for ē_{s} in Lemma 9.10, the expression for ē_{i} in Proposition 9.16, and the equality between the homology classes [L̄_{𝔱,𝔰}^{vir}] and [B̄L_{𝔱,𝔰}^{vir}] in Proposition 10.17 implies that the intersection number in (11.2) can be reduced to pairings of the form

(11.3)  ⟨ν^{a} ⌣ π_{X}^{∗}S^{ℓ}(β) ⌣ π_{𝔰}^{∗}μ_{s}(z_{2}), [B̄L_{𝔱,𝔰}^{vir}]⟩

where β ∈ H^{•}(X; ℝ) and z_{2} ∈ 𝔸_{2}(X).

We wish to compute the pairings (11.3) by writing it as a sum over pairings with the pieces B̄L_{𝔱,𝔰}^{vir}(P_{i}) and then use the fiber bundle structure in Lemma 10.16 of each of these pieces. To do this, we must replace the cohomology classes in (11.3) by cohomology classes with compact support away from the boundaries ∂_{j}B̄L_{𝔱,𝔰}^{vir}(P_{i}). Such a replacement requires a choice and this choice must be done consistently on each piece. We encode these choices in the following geometric data. We define a quotient, B̌L_{𝔱,𝔰}^{vir}, of B̄L_{𝔱,𝔰}^{vir} by replacing the boundaries ∂_{j}B̄L_{𝔱,𝔰}^{vir}(P_{i}) with spaces of codimension greater than or equal to two. The cohomology classes in (11.3) pull back from cohomology classes on this quotient. Because the image of the boundaries in the quotient has codimension greater than or equal to two, we can define cohomology classes with compact support by pulling back cocycles from the quotient. Using these cocycles is equivalent to computing (11.3) using the fundamental class of B̌L_{𝔱,𝔰}^{vir} which, because the image of the boundaries has codimension two, can be written as a sum of fundamental classes of pieces.

In §11.1, we construct the quotient space B̌L_{𝔱,𝔰}^{vir}. This construction shows that the images of each piece B̄L_{𝔱,𝔰}^{vir}(P_{i}) in the quotient has a fiber bundle structure identical to that of B̄L_{𝔱,𝔰}^{vir}(P_{i}) but with a compact fiber and base. In §11.2, we show how to use the quotient to perform the desired computations. Finally in §11.3 we give the proofs of the main theorems of the paper.

11.1. The quotient space. We wish to take advantage of the fiber bundle structure

(11.4)  B̄L_{𝔱,𝔰}^{vir}(P_{j}) → K_{j}

and our control over the structure group of this fibration to compute cohomological pairings with B̄L_{𝔱,𝔰}^{vir}. To do this, we must write the pairing in (11.3) with the homology class [B̄L_{𝔱,𝔰}^{vir}] as a sum over pairings with some kind of homology class representing the space B̄L_{𝔱,𝔰}^{vir}(P_{j}). However, the pieces B̄L_{𝔱,𝔰}^{vir}(P_{j}) have boundaries as described in §10.4 and so would only define relative homology classes. To overcome this, we describe a quotient of B̄L_{𝔱,𝔰}^{vir} obtained by leaving the interiors of the pieces B̄L_{𝔱,𝔰}^{vir}(P_{j}) and replacing the boundaries ∂_{k}B̄L_{𝔱,𝔰}^{vir}(P_{j}) with quotients obtained by deleting part of the gluing data.

To be more precise, for A ⊂ X a topological space and f : A → B a continuous map, we define the compression of X along f to be the quotient space obtained from X by identifying points in A which have the same image under f. The following is an elementary exercise. Lemma 11.3. If A ⊂ X has a neighborhood N_{A} homeomorphic to a mapping cone of some map p : Y → A, then the compression of X along f is homeomorphic to the space obtained by replacing N_{A} with the mapping cone of the map f ∘ p.

We will define the quotient of B̄L_{𝔱,𝔰}^{vir} to be the compressions of B̄L_{𝔱,𝔰}^{vir} along a projection map from ∂_{k}B̄L_{𝔱,𝔰}^{vir}(P_{j}) given by deleting the data in the fiber of ∂_{k}K_{j} → K_{k} (given by (10.53)). Three observations then complete the computation. First, the quotient is the union of the quotients of the pieces, B̌L_{𝔱,𝔰}^{vir}(P_{i}), and the intersection of any two such pieces has codimension-four. Second, the cohomology classes in (11.2) pull back from this quotient and hence the computation can be done on the quotient. Third, the quotient pieces, B̌L_{𝔱,𝔰}^{vir}(P_{i}), all admit the desired fiber bundle structure, (11.4), albeit over a larger space, with the same structure group.

11.1.1. The quotients of K_{i}. We begin by defining a quotient Ǩ_{i} of the compact subspaces K_{j} of Sym^{ℓ}(X) defined in (10.12). We define the quotients inductively. As before, we have enumerated the partitions respecting the ordering of Σ(X^{ℓ}, P_{i}). For j < i, write

(11.5)  π_{j,i} : ∂_{j}K_{i} → K_{j}

for the projection defined by the restriction of π(X^{ℓ}, P_{j}) to the subspace in Lemma 10.21. For k < j < i, the equality

(11.6)  π_{k,i} = π_{k,j} ∘ π_{j,i}

follows from the Thom-Mather equality (4.30) which is shown to hold in Lemmas 4.9-4.11.

If P_{1} is the lowest partition, Ǩ_{1} = K_{1}. For all j with 1 < j, let Ǩ_{j}(1) be the compression of K_{j} along the map π_{1,j} : ∂_{1}K_{j} → K_{1}. Let ∂_{i}Ǩ_{j}(1) be the image of ∂_{i}K_{j} under the projection K_{j} → Ǩ_{j}(1). The equalities

(11.7)  π_{1,j} ∘ π_{j,i} = π_{1,i} for 1 < j < i

imply that the projections π_{j,i} descend to maps on the quotients, π̌_{j,i}(1) : ∂_{i}Ǩ_{i}(1) → Ǩ_{j}(1). Define Ǩ_{2} = Ǩ_{2}(1).

Continue inductively. Assume that

(1) For all i = 1, …, r, a quotient Ǩ_{i}(1, 2, …, p) of K_{i} has been defined,

(2) If i > j and ∂_{j}Ǩ_{i}(1, 2, …, p) is the image of ∂_{j}K_{i} under the quotient map, then the

map π_{j,i} : ∂_{j}K_{i} → K_{j} induces a map
π̌_{j,i}(1, 2, …, p) : ∂_{j}Ǩ_{i}(1, 2, …, p) → Ǩ_{j}(1, 2, …, p),

(3) The quotient Ǩ_{i}(1, 2, …, p) is the compression of Ǩ_{i}(1, 2, …, p − 1) along the map

π̌_{k−1,i}(1, …, p − 1) : ∂_{k−1}K_{i}(1, 2, …, p − 1)

Because the maps π̌_{j,i}(1, 2, …, p) are defined by quotients of the maps π_{j,i}, the analogue of the equality (11.7) holds for i > j > k:

(11.8)  π̌_{k,j}(1, …, p) ∘ π̌_{j,i}(1, …, p) = π̌_{k,i}(1, …, p).

For i > p + 1, we then define Ǩ_{i}(1, …, p + 1) to be the compression of Ǩ_{i}(1, …, p) along the map

π̌_{p+1,i}(1, …, p) : ∂_{p+1}Ǩ_{i}(1, …, p) → K_{p+1}(1, …, p).

For i ≤ p+1, we define Ǩ_{i}(1, …, p+1) = Ǩ_{i}(1, …, p). The equality (11.8) implies that the maps π̌_{j,i}(1, …, p) descend to define maps π̌_{j,i}(1, …, p + 1) on the quotients, completing the induction.

Define Ǩ_{i} = Ǩ_{i}(1, …, i − 1). We have the following characterization of Ǩ_{i}. Lemma 11.4. The quotient Ǩ_{i} is homeomorphic to clΣ(X^{ℓ}, P_{i}).

Proof. The lemma follows from the construction of Ǩ_{i} and from Lemma 11.3. □

11.1.2. The quotients of BL_{𝔱,𝔰}^{vir}(P_{i}). The construction of the quotient B̌L_{𝔱,𝔰}^{vir}(P_{i}) is done in an identical manner to that of Ǩ_{i}. To control the resulting topology, we begin with the following lemma. Lemma 11.5. The fiber bundle B̄L_{𝔱,𝔰}^{vir}(P_{i}) → K_{i} is the pullback of an orbifold fiber bundle B̌L_{𝔱,𝔰}^{vir}(P_{i}) → Ǩ_{i} by the projection map K_{i} → Ǩ_{i}.

Proof. The fiber bundle

Fr(𝔱, 𝔰, P_{i}, g_{P_{i}}) → Δ^{∘}(X^{ℓ}, P_{i})

defined in (6.2) extends from the open diagonal Δ^{∘}(X^{ℓ}, P_{i}) to its closure Δ(X^{ℓ}, P_{i}). The action of the symmetric group, 𝔖(P_{i})/Γ(P_{i}), is no longer free, but only pseudo-free on this extension and hence an orbifold fiber bundle. The quotient space B̌L_{𝔱,𝔰}^{vir}(P_{i}) is defined by using this extension of the above fiber bundle, the identification of Ǩ_{i} with Σ(X^{ℓ}, P_{i}), and leaving the fiber unchanged. □

By the characterization of B̄L_{𝔱,𝔰}^{vir}(P_{i}) as a pullback, there is a map

(11.9)  B̄L_{𝔱,𝔰}^{vir}(P_{i}) → B̌L_{𝔱,𝔰}^{vir}(P_{i})

which we refer to as the quotient map. The following observation about the pseudo-fiber bundle described above is an immediate consequence of its construction. Lemma 11.6. There is a free S^{1} action on the bundle B̌L_{𝔱,𝔰}^{vir}(P_{i}) → Ǩ_{i} defined in Lemma 11.5 such that the quotient map (11.9) is equivariant with respect to the S^{1} action defining the cohomology class ν on B̄L_{𝔱,𝔰}^{vir}(P_{i}) and this S^{1} action on the quotient.

The quotient B̌L_{𝔱,𝔰}^{vir} of B̄L_{𝔱,𝔰}^{vir} is then defined, roughly speaking, by taking the compressions of B̄L_{𝔱,𝔰}^{vir} along the top map in the diagram for i < j

(11.10)

∂_{i}B̄L_{𝔱,𝔰}^{vir}(P_{j}) ⟶^{π_{i,j}^{B}} B̌L_{𝔱,𝔰}^{vir}(P_{j})|Ǩ_{i}
p↓  ↓
∂_{i}K_{j}  ⟶^{π_{i,j}}  ∂_{i}Ǩ_{j}

where we understand ∂_{i}B̄L_{𝔱,𝔰}^{vir}(P_{j}) to be a subspace of B̄L_{𝔱,𝔰}^{vir}.

To carry out the construction of B̌L_{𝔱,𝔰}^{vir}, we use a downwards induction, first taking the compression along the maps π_{i,r}^{B} (where P_{r} is the top stratum). That this may be done for more than one boundary ∂_{j}B̄L_{𝔱,𝔰}^{vir}(P_{r}) (i.e. the map π_{j,r}^{B} descends to the quotient obtained from a compression along π_{i,r}^{B}) follows from the equality (11.6)) on the maps π_{k,i} on ∂_{k}K_{i} and the compatibility of the trivializations of the bundle Fr(𝔱, 𝔰, P_{r}) used to define the maps π_{i,r}^{B}.

To continue the induction, taking the quotient of BL_{𝔱,𝔰}^{vir}(P_{i}), for r > i along maps π_{j,i}^{B}, we must ensure that the maps π_{j,i}^{B} descend to the quotient of BL_{𝔱,𝔰}^{vir}(P_{i}) obtained by compressing along the map π_{i,j}^{B} where i < j. We use the following result describing the compression along the map π_{i,j}^{B}, for i < j, as a compression of a map on a subspace of ∂_{j}B̄L_{𝔱,𝔰}^{vir}(P_{i}).

Lemma 11.7. If i < j and B̄L_{𝔱,𝔰}^{vir}(P_{i}) is compressed along the map

π_{i,j}^{B} : ∂_{i}B̄L_{𝔱,𝔰}^{vir}(P_{j}) → B̌L_{𝔱,𝔰}^{vir}(P_{j})|Ǩ_{i}

then the resulting quotient of B̄L_{𝔱,𝔰}^{vir}(P_{i}) can be described as a fiber bundle,

(11.11)  \begin{aligned}M̌_{j}(P_{i}, ε_{i}) & ⟶ & M̃_{𝔰} ×_{G_{𝔰}×S^{1}} Fr(𝔱, 𝔰, P_{i}, g_{P_{i}}) ×_{G(P_{i})} M̌(P_{i}, ε_{i}) \\ π(𝔱,𝔰,P_{i}) & ↓ \\ & & M_{𝔰} × Ǩ_{i} \end{aligned}

where the fiber M̌_{j}(P_{i}, ε_{i}) is a compression of M̄(P_{i}, ε_{i}) along the map

(11.12)  π_{i,j}^{B,f} : ∂_{j}M̄(P_{i}, ε_{i}) → ∐_{P′′∈[P_{i}<P_{j}]} M̄(P′′, ε_{j})/𝔖(P_{i})

defined by the map (10.57) as described in Lemma 10.25.

Proof. The compression along the map π_{i,j}^{B} is defined by deleting the data in the fiber of the map π_{i,j} in the diagram (11.10). Assume the fiber lies over a point x ∈ ∂_{j}Ǩ_{i} with pre-image in ∂_{k_{1}}∂_{k_{2}} … ∂_{k_{r}}∂_{j}K_{i} and whose pre-image is disjoint from all other corners ∂_{k_{0}}∂_{k_{1}}∂_{k_{2}} … ∂_{k_{r}}∂_{j}K_{i}. Then, the fiber of the map π_{j,i} in the diagram (11.10) given by the fiber of the projection

∂_{k_{1}}∂_{k_{2}} … ∂_{k_{r}}∂_{j}K_{i} → K_{k_{1}}

given by the space ∂_{k_{2}} … ∂_{k_{r}}∂_{j}Z(P_{k_{1}}, [P_{i}], ε) defined in (10.52) (see Lemma 10.53). Using the equality of boundaries,

∂_{i}B̄L_{𝔱,𝔰}^{vir}(P_{j}) = ∂_{j}B̄L_{𝔱,𝔰}^{vir}(P_{i})

and the description of the fiber of ∂_{j}B̄L_{𝔱,𝔰}^{vir}(P_{i}) over x in Lemma 10.25, we see the compression is as described in the lemma. □

Proceeding inductively, one observes that the quotient described in Lemma 11.7 is still the pullback from a bundle (with the same fiber) over Ǩ_{i} because only the fiber has been changed and not the underlying bundle. Thus, one may assume (inductively) that a quotient, B̃L_{𝔱,𝔰}^{vir}(P_{i}), of B̄L_{𝔱,𝔰}^{vir}(P_{i}) has been obtained by all the compressions along ∂_{j}B̄L_{𝔱,𝔰}^{vir}(P_{i}) for i < j as described in Lemma 11.7. Because these compressions only alter the fiber of the bundle B̄L_{𝔱,𝔰}^{vir}(P_{i}) → K_{i}, the quotient B̃L_{𝔱,𝔰}^{vir}(P_{i}) still admits a fiber bundle structure

B̃L_{𝔱,𝔰}^{vir}(P_{i}) → K_{i} which is the pullback of a bundle B̌L_{𝔱,𝔰}^{vir}(P_{i}) as in Lemma 11.5. The bundle

B̌L_{𝔱,𝔰}^{vir}(P_{i}) is a quotient of B̃L_{𝔱,𝔰}^{vir}(P_{i}) compressing along the maps in induced by π_{k,i}^{B} induced on the quotient, allowing us to proceed inductively.

We summarize: Lemma 11.8. Let

F̌r(𝔱, 𝔰, P_{i}, g_{P_{i}}) → Δ(X^{ℓ}, P_{i})

be the extension of the bundle

Fr(𝔱, 𝔰, P_{i}, g_{P_{i}}) → Δ^{∘}(X^{ℓ}, P).

Then, the quotient B̌L_{𝔱,𝔰}^{vir}(P_{i}) of B̄L_{𝔱,𝔰}^{vir}(P_{i}) admits a fiber orbifold bundle structure,

(11.13)  \begin{aligned}M̌(P_{i}) & ⟶ & M̃_{𝔰} ×_{G_{𝔰}×S^{1}} F̌r(𝔱, 𝔰, P_{i}, g_{P_{i}}) ×_{G(P_{i})} M̌(P_{i}, ε_{i}) \\ π(𝔱,𝔰,P_{i}) & ↓ \\ & & M_{𝔰} × clΣ(X^{ℓ}, P_{i}) \end{aligned}

where the fiber M̌(P_{i}) is obtained by compressions of the fiber M̄(P_{i}, ε_{i}) of (10.38) along the maps π_{i,j}^{B,f} defined in (11.12).

The orbifold bundle structure is due to the non-trivial stabilizer of 𝔖(P)/Γ(P) acting on F̄r(𝔱, 𝔰, P) on the points in Δ(X^{ℓ}, P) − Δ^{∘}(X^{ℓ}, P). We then define

(11.14)  B̌L_{𝔱,𝔰}^{vir} = ∪_{i}B̌L_{𝔱,𝔰}^{vir},

to be the quotient of B̄L_{𝔱,𝔰}^{vir} defined by the taking union of the quotients of the pieces as described above. Note that the same quotient is taken on each piece.

Proposition 11.9. The quotient B̌L_{𝔱,𝔰}^{vir} of B̄L_{𝔱,𝔰}^{vir} defined above has the following properties:

(1) The maps π_{X} : B̄L_{𝔱,𝔰}^{vir} → Sym^{ℓ}(X) and π_{𝔰} : B̄L_{𝔱,𝔰}^{vir} → M_{𝔰} descend to the quotient,

(2) There is a cohomology class

(11.15)  ν̌ ∈ H^{2}(B̌L_{𝔱,𝔰}^{vir})

which pulls back to the cohomology class ν ∈ H^{2}(B̄L_{𝔱,𝔰}^{vir}) under the quotient map.

Proof. The first property is immediate from the construction. The second property follows from Lemma 11.6. □

11.1.3. The fundamental classes. We can define a fundamental class

[B̌L_{𝔱,𝔰}^{vir}] ∈ H_{max}(B̌L_{𝔱,𝔰}^{vir})

as was done in (10.35). However, because the intersections of the different pieces B̌L_{𝔱,𝔰}^{vir}(P_{i}) have codimension greater than or equal to two, we can also define fundamental classes,

[B̌L_{𝔱,𝔰}^{vir}(P_{i})] ∈ H_{max}(B̌L_{𝔱,𝔰}^{vir}).

If

π_{B} : B̄L_{𝔱,𝔰}^{vir} → B̌L_{𝔱,𝔰}^{vir}

is the quotient map, then the fundamental classes defined above satisfy:

(11.16)  (π_{B})_{∗}[B̄L_{𝔱,𝔰}^{vir}] = [B̌L_{𝔱,𝔰}^{vir}],

and

(11.17)  [B̌L_{𝔱,𝔰}^{vir}] = ∑_{i} [B̌L_{𝔱,𝔰}^{vir}(P_{i})].

The preceding discussion and the equality ν = π_{B}^{∗}ν̌ in Proposition 11.9 then imply:

Lemma 11.10. For any μ_{𝔰} ∈ H^{•}(M_{𝔰}), S^{ℓ}(β) ∈ H^{•}(Sym^{ℓ}(X)), and z_{2} ∈ 𝔸_{2}(X), the equality

(11.18)

⟨ν^{δ} ⌣ π_{𝔰}^{∗}μ_{𝔰}(z_{2}) ⌣ π_{X}^{∗}S^{ℓ}(β), [BL_{𝔱,𝔰}^{vir}]⟩

= ∑_{i}⟨ν̌^{δ} ⌣ π_{𝔰}^{∗}μ_{𝔰}(z_{2}) ⌣ π_{X}^{∗}S^{ℓ}(β), [B̌L_{𝔱,𝔰}^{vir}]⟩.

11.2. Pullback arguments. The pairings on the right-hand-side of (11.18) can be computed using a pushforward-pullback argument as follows. The pseudo-fiber bundle in (11.11) is branch-covered by the fiber bundle,

(11.19)  \begin{aligned}M̌(P_{i}, ε_{i}) & ⟶ & M̃_{𝔰} ×_{G_{𝔰}×S^{1}} Fr(𝔱, 𝔰, P_{i}, g_{P_{i}}) ×_{G̃(P_{i})} M̌(P_{i}, ε_{i}) \\ π(𝔱,𝔰,P_{i}) & ↓ \\ & & M_{𝔰} × Δ(X^{ℓ}, P_{i}) \end{aligned}

Let B̂L_{𝔱,𝔰}^{vir}(P_{i}) denote the fiber bundle in (11.19). (The actual degree of the covering, |𝔖(P_{i})/Γ(P_{i})|, is not relevant to this discussion.) Hence, we have

(11.20)  ⟨ν̌^{δ} ⌣ π_{𝔰}^{∗}μ_{𝔰} ⌣ π_{X}^{∗}S^{ℓ}(β), [B̌L_{𝔱,𝔰}^{vir}]⟩ = q_{i}⟨ν̌^{δ} ⌣ π_{𝔰}^{∗}μ_{𝔰} ⌣ π_{X}^{∗}S^{ℓ}(β), [B̂L_{𝔱,𝔰}^{vir}]⟩

for some rational number q. Let G_{i} = G̃(P_{i}). Let f_{𝔱,𝔰,i} : Δ(X^{ℓ}, P_{i}) → BG_{i} be the classifying map. We then have a diagram,

(11.21)

B̌L_{𝔱,𝔰}^{vir}  ⟶^{f̃_{𝔱,𝔰,i}} S^{1}\ EG_{i} ×_{G̃(P_{i})}M̌(P_{i}, ε_{i})
π(𝔱,𝔰,P_{i})↓  ↓
Δ(X^{ℓ}, P_{i}) ⟶^{f_{𝔱,𝔰,i}}  BG_{i}

Observe that ν̌ is the pullback of first Chern class of the S^{1} action on EG_{i} ×_{G̃(P_{i})}M̌(P_{i}, ε_{i}). Then, a pushforward pullback argument applied to the diagram (11.21) yields Lemma 11.11. There are universal polynomials, m_{ℓ,δ,𝔱,𝔰,i} ∈ H^{•}(M_{𝔰} × Δ(X^{ℓ}, P_{i})), in the characteristic classes of the spin^{u} structure V_{ℓ} and the cohomology classes μ_{𝔰}(·) such that

(11.22)  \begin{matrix}⟨ν̌^{δ} ⌣ π_{𝔰}^{∗}μ_{𝔰}(z_{2}) ⌣ π_{X}^{∗}S^{ℓ}(β), [B̂L_{𝔱,𝔰}^{vir}]⟩ \\ = ⟨m_{ℓ,δ,𝔱,𝔰,i} ⌣ μ_{𝔰}(z_{2}) ⌣ π_{X}^{∗}S^{ℓ}(β), [M_{𝔰} × Δ(X^{ℓ}, P_{i})]⟩. \end{matrix}

11.3. Proof of the main theorem. We now prove Theorems 11.1, 1.1, and 11.2.

Proof of Theorem 11.1. The discussion prior to (11.3), Lemma 11.10, and Lemma 11.11 imply that the pairing on the left-hand-side of (11.1) can be reduced to an algebraic expression in terms of the form appearing on the right-hand-side of (11.22). The result then follows from a discussion on the type of terms which will appear. First, observe that factors of Q_{X}(h) will arise only from products of S^{ℓ}(h) ⌣ S^{ℓ}(h). There are at most (δ − 2m)/2 of these factors. In addition, the dimension of the highest stratum in Sym^{ℓ}(X) is 4ℓ, so we can have no power of S^{ℓ}(h) ⌣ S^{ℓ}(h) greater than ℓ. Thus, the power of Q_{X}(h) appearing in the formula must be less than min[(δ − 2m)/2, ℓ].

The factors of ⟨c_{1}(𝔰) − c_{1}(𝔱), h⟩ and ⟨c_{1}(𝔱), h⟩ appearing in (11.1) can be understood by the algebraic work in [11, §6.1]. □

Proof of Theorem 11.2. The condition that ⟨μ_{𝔰}(z_{2}), [M_{𝔰}]⟩ = 0 for all z_{2} ∈ 𝔸_{2}(X) and the expression on the right-hand-side of (11.22), (to which the intersection number can be reduced) implies that all pairings will be multiples of SW_{X,𝔰}(·). □

Proof of Theorem 1.1. Theorem 1.1 follows from Theorem 11.1 and the cobordism formula (10.11). The polynomials p_{δ,ℓ,m,i} appearing in Theorem 1.1 differ from the polynomials q_{δ,ℓ,m,i} appearing in Theorem 11.1 by the factors of two appearing in (2.52) and factors of negative one from the change of orientation expression in (10.8) as derived in the analogous formula in [11] (see [11, Theorem 1.4]). □

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Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019

E-mail address: feehan@math.rutgers.edu
URL: http://www.math.rutgers.edu/∼feehan

Department of Mathematics, Florida International University, Miami, FL 33199

E-mail address: lenesst@fiu.edu
URL: http://www.fiu.edu/∼lenesst