Math 527: Homotopy Theory

Spring 2011

Bert Guillou

• From MIT's Algebraic Topology student seminar, two lists (here and here) of papers in the subject.
• My old Category Theory notes
• The UIUC Topology seminar calendar
• Free online texts: Hatcher and May
• Neil Strickland's notes on compactly generated spaces

• Week 1. Homotopy, homotopy equivalence, the categories of based and unbased space.
• Week 2. Higher homotopy groups, weak homotopy equivalence, CW complex.
• Week 3. Cofibrations and the Homotopy Extension Property.
• Week 4. Relative homotopy groups, homotopy fiber, long exact sequence in homotopy, Whitehead theorem.
• Week 5. Cellular and CW approximation, the homotopy category, cofiber sequences.
• Week 6. Homotopy pushouts, fibrations and the Homotopy Lifting Property, Serre fibrations.
• Week 7. Homotopy pullbacks, Homotopy Excision, Freudenthal suspension theorem.
• Week 8. Stable homotopy groups, Hurewicz theorem, homology Whitehead theorem.
• Week 9. Proof of Blakers-Massey, Eilenberg-Mac Lane spaces.
• Week 10. Constructions of Eilenberg-Mac Lane spaces, representation of cohomology by K(A,n)'s, obstruction theory.
• Week 11. Postnikov towers, principal fibrations.
• Week 12. Localization, finiteness of homotopy groups of spheres.

• Homework 1 (due Friday, Jan. 28)
• Homework 2 (due Friday, Feb. 11)
• Homework 3 (due Friday, Feb. 25)
• Homework 4 (due Wednesday, Mar. 16)
• Homework 5 (due Wednesday, Apr. 13).