Algebraic Topology

  • Instructor: Dr. András STIPSICZ
  • Contact: stipsicz dot andras at renyi dot hu
  • Prerequisites: Basic algebra: vector spaces, groups, factor groups, homomorphisms.
    Basic analysis in \R^n: continuous maps, convergence, differentiable maps.
  • Text:  Handouts and Allan Hatcher: Algebraic Topology
    Additional reference: Fuchs - Fomenko - Gutenmacher: Homotopic topology; Husemoller: Fiber bundles

Course 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:

  1. Simplicial and singular homology
  2. Basic homological algebra (chains and homotopies)
  3. Degree, CW-homology
  4. Cohomology, ring structure
  5. Orientability, Poincare duality
  6. Obstruction theory
  7. Fiber bundles, principal bundles
  8. Classification of vector bundles
  9. Characteristic classes
  10. Knots and knot invariants
  11. The Jones polynomial and Khovanov homology