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


[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


[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.