MATH 748R, Homotopy Theory

Instructor:

Jonathan Rosenberg. You may call me at (301) 405-5166 or reach me by email at jmr@math.umd.edu. I usually attend the Geometry/Topology Seminar (Mondays 3-4), the Representation Theory and Algebra Seminars (Mondays and Wednesdays 2-3), and the Colloquium. My office hours are Mondays and Wednesdays after class, or by appointment.

Meetings:

MWF at 9 in MTH 0102.

Prerequisites:

MATH 730 and 734 or equivalent.

Texts:

These texts are available free on the web. If you have another standard algebraic topology book such as Bredon or Spanier, that will also be useful.

  1. A Concise Course in Algebraic Topology by J. Peter May, Chicago Lectures in Math., Univ. of Chicago Press, 1999. Available here.
  2. Algebraic Topology by Allen Hatcher, Cambridge University Press, 2002, Chapter 4, available here.
  3. Vector Bundles and K-Theory and Spectral Sequences in Algebraic Topology by Allen Hatcher, still incomplete, but available here.

Another useful reference (too bulky to use as a text, however) is George W. Whitehead, Elements of Homotopy Theory, Springer, 1978.

Quick Summary:

This course will attempt to bridge the (unfortunately large) gap between what is covered in MATH 734 and the topology one really needs for research in geometry and topology (or even some aspects of analysis). Topics will include: higher homotopy groups, the Hurewicz theorem, fiber bundles and fibrations, methods for computing homotopy and homology groups, vector bundles, and characteristic classes. There will be no exams, but regular problem sets will be assigned.

Course Evaluations

Please go to Course Eval UM to fill in your course evaluation before Friday, May 11.

Schedule (will be updated as the semester progresses):

Week Topic Reading Assignment Notes
1/23-1/27 Introduction, language, foundations M, § 2.1-2.6, Ch. 5 Classes start W, 1/25. Homework 1 due F, 2/3.
1/30-2/3 Homotopy groups, cofibrations M, Ch. 6; Steenrod's paper on NDR pairs  
2/6-2/10 Fibrations and applications of the exact sequence M, Ch. 7, 9 Homework 2 due F, 2/17.
2/13-2/17 CW complexes and Whitehead's Theorem M, Ch. 10; H-AT, second half of 4.1 Homework 3 due F, 2/24.
2/20-2/24 The homotopy excision and the Hurewicz theorems M, Ch. 11 and 15; H-AT, § 4.2  
2/27-3/2 The homotopy excision and the Hurewicz theorems (cont'd) M, Ch. 11 and 15; H-AT, § 4.2 Homework 4 due F, 3/9.
3/5-3/9 Eilenberg-MacLane spaces and beginnings of obstruction theory M, § 16.4, 16.5, 18.5; H-AT, § 4.3, pp. 410-419 Homework 5 due F, 3/30
3/12-3/16 Postnikov systems, obstruction theory (cont'd) M, § 22.2, 22.4; H-AT, § 4.3, pp. 410-419 no class Friday 3/16
3/19-3/23 Spring Break, No Class
3/26-3/30 The Serre spectral sequence H-SS, Ch. 1 Solutions to Homework 5
4/2-4/6 The Serre spectral sequence: more details and applications H-SS, Ch. 1 Homework 6 due M, 4/16
4/9-4/13 Serre classes and applications H-SS, Ch. 1 no class Friday 4/13,
Solutions to Homework 6
4/16-4/20 Serre classes and rational homotopy groups H-SS, Ch. 1 Homework 7 due F, 4/27
4/23-4/27 Hopf algebras and cohomology rings of Lie groups and classifying spaces H-AT, § 3.C  
4/30-5/4 Vector bundles and characteristic classes M, § 23.1; H-VB, § 1.1-1.2 Solutions to Homework 7.
Homework 8, due M, 5/14
5/7-5/11 Vector bundles and characteristic classes (cont'd)   W, 5/9 is last day of class.
Solutions to Homework 8.
Under "Reading Assignment", M denotes May's book; H-AT denotes Hatcher's book "Algebraic Topology"; H-VB denotes Hatcher's book "Vector Bundles and K-Theory"; H-SS denotes Hatcher's book "Spectral Sequences in Algebraic Topology".