MATH 7221 — Topology 2

Course description Welcome to MATH 7221. This course will cover the fundamental computational devices in algebraic topology (the fundamental group, homology and cohomology), as well as their applications to the study of topological spaces (e.g., fixed-point theory, invariance of domain, the Borsuk-Ulam theorem, etc). Prerequisites include point-set topology (e.g., connectedness, compactness, continuity, etc) and abstract algebra (e.g., groups, rings, fields and their homomorphisms).

Syllabus: pdf

Topics:

  • The fundamental group and its computation via covering spaces

  • Cellular and simplicial complexes

  • Chain complexes and basic homological algebra

  • Homology and applications (e.g., Brouwer’s fixed-point theorem, invariance of domain theorem)

  • Cohomology

  • Cup products and applications (e.g., the Borsuk-Ulam theorem)

Homework:

  • Homework 1 pdf | Due 10/05/2026

  • Homework 2 pdf | Due 10/26/2026

Lectures:

  • Lecture 1 [09/11/2026] pdf: Course logistics, intro to algebraic topology

  • Lecture 2 [09/15/2026] pdf: The fundamental group, homomorphisms induced by maps and change of basepoint

  • Lecture 3 [09/18/2026] pdf: Deformation retractions, fundamental group of the circle implies Brouwer’s fixed point thm and the fundamental thm of algebra

  • Lecture 4 [09/22/2026] pdf: Higher dimensional spheres are simply-connected, pushouts and the Seifert-Van Kampen theorem

  • Lecture 5 [09/25/2026] pdf: Covering spaces, lifting properties, the fundamental group of the circle

  • Lecture 6 [09/29/2026] pdf: The lifting criterion for maps, the Galois correspondence

  • Lecture 7 [10/06/2026] pdf: The universal cover

  • Lecture 8 [10/09/2026] pdf: The group of Deck transformations, the action of the fundamental group

  • Lecture 9 [10/13/2026] pdf: Singular homology

  • Lecture 10 [10/16/2026] pdf: H0 measures path-connected components and H1 holes (the Hurewicz theorem), functoriality of homology

  • Lecture 11 [10/20/2026] pdf: Delta complexes, abstract simplicial complexes and simplicial homology

  • Lecture 12 [10/23/2026] pdf: Chain complexes, chain maps, chain homotopies and homotopy invariance of homology

  • Lecture 13 [10/27/2026] [pdf: Exact sequences (pair, triple), homology of spheres, Invariance of Domain