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:
*S*,^{n}**R***P*,^{n}**C***P*,^{n}*T*,^{n}*S*^{2}^*S*^{3}, 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
- π
_{1}(*X*,*x*_{0}) - Van Kampen’s theorem
- Higher homotopy groups and the Hurewicz theorem
- π
_{3}(*S*^{2}) - Higher homotopy groups of the sphere

III Covering Spaces

- Definition of a covering projection
- Examples — Coverings of
*S*^{1},*S*covering^{n}**R***P*,^{n}*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