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Algebraic Topology

Learn to state and prove basic theorems in algebraic topology, and compute the (co)homology of topological spaces and interpret the results geometrically.

The course covers:

  • singular homology and cohomology of topological spaces
  • exact sequences, chain complexes and homology
  • homotopy invariance of singular homology
  • the Mayer-Vietoris sequence and excision
  • cell complexes and cellular homology
  • the cohomology ring
  • homology and cohomology of spheres and projective spaces
  • applications such as the Brouwer Fixed Point theorem, the Borsuk-Ulam theorem and theorems about vectorfields on spheres

This course is given jointly by Stockholm University and KTH, and can be a part of the Master's Programme in Mathematics but may also be taken as a separate course.