Algebraic Topology and Homological Algebra — ATH

  • Instructor: Ádám Gyenge
  • Contact: {gyenge.adam} (at) ttk [dot] bme [dot] hu
  • Prerequisites: Basic algebra: vector spaces, groups, factor groups, homomorphisms, rings and ideals.
    Basic topology: topological spaces, continuous maps, homeomorphisms, homotopy, constructions.
    Here is a summary/review of basic topology (from a course at the Queen Mary University, London).
  • Text: Allen Hatcher: Algebraic Topology
    Joseph Rotman: Introduction to Homological Algebra

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 and algebra. Ideas from homology are present in all modern directions of mathematics, and we will show several appearances of those as well.

Topics:

  • Simplicial and singular homology
  • Basic homological algebra (chains and homotopies, categories and functors)
  • Degree, CW-homology
  • Cohomology, ring structure
  • Orientability, Poincare duality
  • Obstruction theory
  • Fiber bundles, principal bundles
  • Classification of vector bundles
  • Characteristic classes
  • Advanced homological algebra, Hom, Tensor
  • Derived functors

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