Algebraic Topology — ALT

  • Instructor: Clayton McDonald
  • Contact: claytkm 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 course is to provide an introduction to basic notions in algebraic topology (homotopy, homology, and cohomology) and show some simple (and potentially more complicated) applications of these techniques. We will use primarily examples in knot theory and low dimensional topology as guiding examples.

Topics covered:

  • the fundamental group and covering spaces
  • simplicial and singular homology
  • basic homological algebra (chains, homotopies, exact sequences)
  • cohomology and the cup product
  • Poincare duality and the Cap product
  • knots and knot invariants (the Alexander polynomial and Jones polynomial - time permitting)
  • CW complexes and various constructions

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