Oral Qualifying Exam Syllabus

Algebraic Topology and Cobordism Theory

Algebraic Topology

1. The Fundamental Group
   • Equivalent definitions of \(\pi_1(X)\)
   • The van Kampen theorem
   • Computing the fundamental group of CW complexes
   • Covering spaces, fundamental theorem
2. Homology
   • Simplicial, singular, and cellular homology
   • Simplicial approximation
   • Homology theories
   • Classical applications
     1. Brouwer fixed-point theorem
     2. Jordan curve theorem and generalization
     3. \(\pi_1(X)_{ab} \cong H_1(X) \)
     4. Borsuk-Ulam Theorem
     5. Invariance of dimension
     6. Invariance of domain
     7. Hairy ball theorem
     8. Degree theory
     9. Lefschetz fixed-point theorem
3. Cohomology
   • Simplicial, singular, cellular, de Rham cohomology
   • Cohomology with local coefficients, presheaf cohomology
   • Cup, cap, and cross products
   • Thom isomorphism theorem
   • Künneth Formulas
   • Poincaré duality, Alexander duality, Lefschetz duality
   • Universal Coefficients for Homology
   • Cohomology theories
4. Homotopy Theory
   • Whitehead's theorem
   • Cellular approximation theorem, CW approximation theorem
   • The Hurewicz theorem
   • Fibrations, HELP, long exact sequence
   • Homotopy groups of spheres, stable structure, Freudenthal suspension theorem
   • Homotopy construction of cohomology
5. Generalized cohomology and homology theories
   • Spectra, Eilenberg MacLane spectrum
   • The Brown Representability Theorem
   • Steenrod algebra, cohomology operations

Cobordism Theory

1. \(h\)-cobordism theorem
2. \( (B,f) \) manifolds and cobordisms
3. Computation of \(MO_*\), \(MSO_*\), \(MU_*\)
4. Stiefel-Whitney classes, Chern classes, Pontrjagin classes
5. Oriented cohomology theories

References

[DavKir]   Davis, J.; Kirk, P., Lecture Notes in Algebraic Topology  
[HatAT]    Hatcher, A., Algebraic Topology.
[BotTu]    Bott, R.; Tu, L., Differential Forms in Algebraic Topology.
[Vic]      Vick, J., Homology Theory.
[MayAT]    May, J. P., A Concise Course in Algebraic Topology.
[Wes]      Weston, T., An Introduction to Cobordism Theory.
[MilSta]   Milnor, J.; Stasheff, J., Characteristic Classes.
[Sto]      Stong, R., Notes on Cobordism Theory.
[Ada]      Adams, J. F., Stable Homotopy and Generalised Homology.

Homological Algebra and Category Theory

Homological Algebra

1. Chain complexes of modules
   • Mapping cones and cylinders
2. Derived functors
   • Injective and projective objects for certain Abelian categories
   • Injective and projective resolutions for certain Abelian categories
   • \( \mathrm{Tor} \), \( \mathrm{Ext} \), \( \lim^1 \),
   • Mittag-Leffler condition
   • Acyclic assembly lemma
3. Universal coefficient theorems
4. Spectral sequences
   • Serre-Leray spectral sequence for fibrations, ordinary homology and cohomology
   • Gysin sequence, Wang sequence
   • Exact sequence of low dimensional terms
   • Serre exact sequence
   • Computation of \( H^{*} \Omega ( \mathbb{S}^n ) \), \( \pi_4 (\mathbb{S}^3) \), \( \pi_4 (\mathbb{S}^2) \)
   • Atiyah-Hirzebruch Spectral sequence
   • K-theory computations using the AHSS
   • Computations with \( MSO_* \)
   • Exact couples and derivation of Serre-Leray spectral sequence
   • Grothendieck spectral sequence

Category Theory

1. Adjoint functors, adjoint functor theorems
   • Adjoints and exactness of functors
2. (Co-)limits
   • Filtered limits and colimits
   • (co-)homology of (co-)limits
3. Abelian categories
   • Examples and properties: \(R\mathrm{-\underline{Mod}}\), \(\mathrm{\underline{Sheaves}}(X)\), \(\mathrm{\underline{Presheaves}}(X)\), \(\mathrm{\underline{Ch}}\), functor categories \(\mathcal{A}^{\mathcal{C}}\)
   • Freyd-Mitchell Embedding Theorem
4. Simplicial sets
   • Kan fibrations
   • Yoneda lemma
   • Geometric realization
5. Monoidal categories
6. Model categories

References

[HS]     Hilton, P.; Stammbach, U., A Course in Homological Algebra.
[Hov]    Hovey; Shipley; Smith, Symmetric Spectra.  
[MacCW]  Mac Lane, S., Categories for the Working Mathematician.
[Wei]    Weibel, C., An Introduction to Homological Algebra.
[MaySO]  May, J. P., Simplicial Objects in Algebraic Topology.
[Mit]    Mitchell, B., Theory of Categories. 

---