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
