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Algebraic Topology

This course is taken in sequence, part 1 in the fall, and part 2 in the spring.

Algebraic Topology I

I Homology Theory

  • Singular homology — definition, simple computations
  • Cellular homology — definition
  • Eilenberg-Steenrod Axioms for homology
  • Computations: Sn, RPn, CPn, Tn, S2^S3, Grassmannians, X*Y
  • Alexander duality — Jordan curve theorem and higher dimensional analogues
  • Applications: Winding number, degree of maps, Brouwer fixed point theorem
  • Lefschetz fixed point theorem

II Homotopy Theory

  • Homotopy of maps, of pointed maps
  • The homotopy category and homotopy functors –examples
  • π1(X, x0)
  • Van Kampen’s theorem
  • Higher homotopy groups and the Hurewicz theorem
  • π3(S2)
  • Higher homotopy groups of the sphere

III Covering Spaces

  • Definition of a covering projection
  • Examples — Coverings of S1, Sn covering RPn, Spin(n) covering SO(n)
  • Homotopy path lifting
  • Classification of coverings of a reasonable space

IV Homology with Local Coefficients

  • Local coefficient systems
  • Relation with covering spaces
  • Obstruction theory
  • The Alexander polynomial of a knot

Algebraic Topology II

I Cohomology

  • Cup products
  • Pairings homology
  • Cohomology and homology with coefficients
  • Universal coefficient theorems

II Cech Cohomology

  • Open coverings and Cech cochains
  • The coboundary mapping
  • Cech cohomology
  • Comparison with singular cohomology

III Selected Topics

  • Group Cohomology
  • Sheaf Cohomology
  • de Rham’s theorem
  • Morse functions and Poincaré duality for manifolds
  • Thom Isomorphism Theorem and cohomology classes Poincaré dual to cycles
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