Math 656: Algebraic Topology

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Catalog Information


Algebraic Topology.

Credit Hours



Math 371 (or equivalent) and Math 553.


A rigorous treatment of the fundamentals of algebraic topology, including homotopy (fundamental group and higher homotopy groups) and homology and cohomology of spaces.

Desired Learning Outcomes


Topology of manifolds, tensors, and orientation is assumed from Math 655. A basic knowledge of fundamental groups and covering spaces, as in Math 554, will also be required.

Minimal learning outcomes

Outlined below are topics that all successful Math 656 students should understand well. Students should be able to demonstrate mastery of relevant vocabulary, and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations.

  1. Fundamental group and homotopy
    • Constructions
    • Van Kampen Theorem
    • Covering spaces and group actions
    • Higher homotopy groups
  2. Homology
    • Simplicial, singular, cellular
    • Exact sequences and excision
    • Mayer-Vietoris sequences
    • Homology with coefficients
    • Homology and the fundamental group
  3. Cohomology
    • Universal coefficient theorem
    • Cup product
    • Poincare duality

Additional topics

At the discretion of the instructor as time allows. Possible topics include de Rham cohomology and the de Rham theorem, additional duality theorems, including Alexander and Lefschetz duality, additional homotopy theory, including Whitehead's theorem, the Hurewicz theorem, fiber bundles, fibrations, obstruction theory.

Courses for which this course is prerequisite

Math 751R