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

  • Lecture 14 [03/10/2026] pdf: The Freudenthal suspension theorem, homotopy excision (the Blakers-Massey theorem part I)

  • Lecture 15 [03/13/2026] pdf: Proof of the Blakers-Massey theorem

  • Lecture 16 [03/17/2026] pdf: The mapping path space and the homotopy fiber of a map. What is a spectral sequence?

  • Lecture 17 [03/24/2026] pdf: Exact couples, derived couple, the exact couple of a filtration

  • Lecture 18 [03/27/2026] pdf: Convergence of the filtration spectral sequence and ensuing extension problems

  • Lecture 19 [03/31/2026] pdf: Towards the Serre Spectral Sequence (SSS)

  • Lecture 20 [03/04/2026] pdf: The proof of the SSS, application (the homology of the loop space of the sphere)

  • Lecture 21 [04/07/2026] pdf: Fibrations with spheres, the Hopf inviariant 1 problem, the cohomology SSS, cohomology ring for the loop space of spheres

  • Lecture 22 [04/10/2026] pdf: Serre classes, pi_{n+1} (S^n) is Z/2 for n > 2 .