Algebraic Topology — ALT

  • Instructor: András Csépai
  • Contact: csepai.andras@renyi.hu
  • Prerequisites: Background in algebra including vector spaces, groups, factor groups, homomorphisms. Background in analysis in R^n: continuous maps, convergence, differentiable maps. Familiarity with point-set topology.
  • Text:
    Alan Hatcher: Algebraic topology;
    Fomenko, A. T.; Fukes, D. B.; Gutenmacher, V. L.: Homotopic topology.

Course description:

The goal of this course is to provide an introduction to the basic notions of algebraic topology (homotopy, homology and cohomology), and to show some simple (and some more sophisticated) applications of these techniques. The main idea is that algebraic objects can be used to describe various properties of topological (and geometric) objects, hence studying the algebra yields information in topology and geometry. We will illustrate this on examples in the theories of manifolds, vector bundles and knots.

Topics (tentative):

  1. the fundamental group and covering spaces
  2. simplicial and singular homology
  3. basic homological algebra
  4. cohomology and the cup product
  5. manifolds
  6. orientability and the Poincaré duality
  7. degree and CW homology
  8. homotopy groups
  9. obstruction theory
  10. fibre bundles and principal bundles
  11. classification of vector bundles
  12. cobordism and the Pontryagin--Thom construction
  13. knots and knot invariants.

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