Algebraic Topology

  • Instructor: dr. Boldizsar Kalmar
  • Contact: boldizsar.kalmar at gmail dot com
  • Prerequisites: Basic algebra: vector spaces, groups, factor groups, homomorphisms. Basic analysis in R^n: continuous maps, convergence, differentiable maps.
  • Text: Allan Hatcher: Algebraic Topology and class notes

Course description: The goal of the lectures is to give a wide and detailed view of the most important tools in algebraic and geometric topology. These connect the most natural "naive" and elementary geometric ideas and constructions with various fields of modern mathematics. We will see how deeply algebra and geometry interact in the study of a mathematical problem. The course is useful to students who want to have a global picture of a big part of today's mathematics or to students who are interested in mathematical research.

Topics covered:

  1. CW complexes in Euclidean spaces, surfaces, manifolds, constructions: We construct topological spaces in R^n, which are important in practice. We give many examples and study their basic properties.
  2. Homology groups and functors: We define homology groups and see how to use them in solving topological problems.
  3. Exact sequences: We learn an abstract algebraic tool to compute the homology groups and see many applications.
  4. Homotopy groups: We study a natural generalization of fundamental group, which helps to classify continuous maps.
  5. Knots, knot invariants, fundamental group, Jones polynomial: We study knots in R^3, and see how to work with their basic invariants.
  6. Morse functions, classification of surfaces: We see a geometric proof for the classification of surfaces. We extend real analysis to surfaces and manifolds.