*Instructor:* Dr. András STIPSICZ

*Text:* Handouts and
Allan Hatcher: Algebraic Topology

Additional reference:
Fuchs - Fomenko - Gutenmacher: Homotopic topology;
Husemoller: Fiber bundles

*Prerequisites:*

Basic algebra: vector spaces, groups, factor groups, homomorphisms.

Basic analysis in \R^n: continuous maps, convergence, differentiable maps.

*Description:*

The goal of the course is to provide an introduction to the basic
notions of homology and cohomology theory, and show some simple (and
some more sophisticated) applications of these techniques in topology.
Ideas from homology are present in all modern directions of
mathematics, and we will show some appearances of those as well. If
time premits, at the end of the course we will see further
applications of the concept of homology in knot theory.

*Topics covered:*

- Simplicial and singular homology
- Basic homological algebra (chains and homotopies)
- Degree, CW-homology
- Cohomology, ring structure
- Orientability, Poincare duality
- Obstruction theory
- Fiber bundles, principal bundles
- Classification of vector bundles
- Characteristic classes
- Knots and knot invariants
- The Jones polynomial and Khovanov homology