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
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, Part I
Lecture 7 [10/06/2026] pdf: The universal cover