MATH 7321 — Topology 3
Course description Welcome to MATH 7321. This course is a continuation of Topology 2 (the Fundamental group, Homology and Cohomology) and introduces tools from homotopy theory such as homotopy groups and fibrations, as well as computational techniques like spectral sequences.
Syllabus: pdf
Topics:
Homotopy groups
Fibrations and fiber bundles
The theorems of Whitehead, Hurewicz and Dold-Thom
Eilenberg-MacLane spaces and the representability of cohomology
Cohomology operations
The theorems of Blakers-Massey and Freudenthal
Stable homotopy groups
Spectral sequences
Lectures:
Lecture 1 [01/09/2026] pdf: Course overview, homotopy groups
Lecture 2 [01/13/2026] pdf: homotopy groups as functors, the homotopy groups of the circle
Lecture 3 [01/16/2026] pdf: Whitney, Sard and low-dimensional homotopy groups of spheres/projective spaces
Lecture 4 [01/20/2026] pdf: Existence of K(G,1)’s (G Abelian and fin gen), long exact seq of the pair, Fiber bundles
Lecture 5 [01/23/2026] pdf: Homotopy Lifting Property (HLP), homotopy long exact sequence for fiber bundles
Lecture 6 [01/27/2026] pdf: Fibrations and their homotopy LES, homotpy groups of the (based) loop space
Lecture 7 [01/30/2026] pdf: Homotopy Extension Property (HEP), discriminative power of homotpy groups, towards Whitehead’s thm
Lecture 8 [02/03/2026] pdf: Proof of Whitehead’s thm, weak homotopy equivs induce (co)homology isomorphisms
Lecture 9 [02/06/2026] pdf: The Hurewicz homomorphism/theorem, Eilenberg-Blakers homology
Lecture 10 [02/10/2026] pdf: The proof of Hurewicz, infinite symmetric products and the Dold-Thom theorem
Lecture 11 [02/13/2026] pdf: The Milnor construction and the existence of Eilenberg-MacLane spaces
Lecture 12 [02/17/2026] pdf: The representability of cellular cohomology and uniqueness of Eilenberg-MacLane spaces
Lecture 13 [02/20/2026] pdf: Cohomology operations (change of coefficients, Bockstein, cup square, Steenrod squares) and their classification