Syllabus Math 525


Math 525. Algebraic Topology I Instructor Syllabus

Fundamental group and covering spaces [first half of semester]

  • Definition of the fundamental group.
  • Covering spaces and lifts of maps.
  • Computing the fundamental group via covering spaces.
  • Applications, such as the Fundamental Theorem of Algebra and the Brouwer fixed point theorem in 2d.
  • Deforming spaces: retraction and homotopy equivalence.
  • Quotient topology and cell complexes.
  • Homotopy extension property and applications to homotopy equivalence.
  • Fundamental groups of CW complexes.
  • Van Kampen's Theorem.
  • Covering spaces and subgroups of the fundamental group.
  • Universal covers.
  • The definitive lifting criterion, classification of covering spaces.
  • Covering transformations and regular covers.

Homology [second half of semester]

  • Delta complexes and their cellular homology.
  • Singular homology.
  • Homotopic maps and homology.
  • The long exact sequence of the pair.
  • Relative homology and excision.
  • Equality of cellular and singular homology.
  • Applications, such as degree of maps of spheres, invariance of dimension, and the Brouwer fixed point theorem.
  • Homology of CW complexes.
  • Homology and the fundamental group: the Hurewicz theorem.
  • Euler characteristic.
  • Homology with coefficients.
  • Intro to categories and axiomatic characterization of homology theories.
  • Further applications, such as the Jordan curve theorem, wild spheres, invariance of domain.

Examples of more detailed pacings for this course can be found at


All of this material may be covered on the 525 comp exam.

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. Freely downloadable at
The content of the course is essentially Chapters 0-2.