Topologie algébrique

MATH 4202: Algebraic topology

Prerequisites: basic notions of general topology (topological spaces, homotopy and fundamental group; we will quickly recap these notions at the beginning of the course)

Program:

Part 1: HOMOLOGY (12 hours)

- CW complexes

- Singular and simplicial homology

- Basic properties of singular homology: homotopy invariance, exact sequences, excision

- Methods of computation: cellular homology, Mayer-Vietories sequence, Künneth formula

- Applications: topological degree, hairy ball theorem, Brower fixed point theorem

Part 2: COHOMOLOGY (12 hours) - Singular cohomology

- Cup product

- Universal coefficient theorem

- De Rham cohomology

- Poincaré and Alexandre dualities - Gysin sequences

References:

- Hatcher: Algebraic Topology

- Bott & Tu: Differential Forms in Algebraic Topology