# Math 527: Homotopy Theory

# Spring 2011

### Bert Guillou

**email:**bertg ♠ illinois ♦ edu

**Phone:**(217) 244-0286

**Office:**330 Illini Hall

**Time & Location:**MWF 1:00-1:50, 441 Altgeld.

**Office Hours:**Thursday, 3-4:00.

**Links:**

- 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

**Lecture Notes:**

- 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:**

- 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).

Department of Mathematics College of Liberal Arts and Sciences University of Illinois at Urbana-Champaign 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Department Main Office Telephone: (217) 333-3350 Fax (217) 333-9576 |