MATH 6510 - Algebraic Topology

Tim Riley, spring 2017.

This course provides an introductory study of certain processes for associating algebraic objects such as groups to topological spaces. The main topics are:

1. The Fundamental Group. Definition and basic properties. The fundamental group of the circle. Van Kampen's theorem. Further calculations and applications.
2. Covering Spaces. Lifting properties. The universal cover. The classification theorem (Galois correspondence). Deck transformations and group actions.
3. Homology Theory. Definitions of simplicial and singular homology. Homotopy invariance. Exact sequences and excision. Degree of maps. Cellular homology of CW complexes. Mayer-Vietoris sequences. Applications. The language of categories and functors. Axioms for homology.
4. A brief introduction to cohomology.

Textbook

Allen Hatcher, Algebraic Topology.