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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.

  • Course structure

    The course consists of one element.

    Teaching format

    Instruction consists of lectures and exercises.


    The course is assessed through written assignments and oral presentations of the assignments.


    A list of examiners can be found on

    Exam information

  • Schedule

    The schedule will be available no later than one month before the start of the course. We do not recommend print-outs as changes can occur. At the start of the course, your department will advise where you can find your schedule during the course.
  • Course literature

    Note that the course literature can be changed up to two months before the start of the course.

    A. Hatcher: Algebraic topology. Cambridge University Press.

    List of course literature Department of Mathematics

  • More information

    New student
    During your studies

    Course web

    We do not use Athena, you can find our course webpages on

  • Contact