MATH 748H, Introduction to Homotopy Theory & Characteristic Classes

Instructor:

Jonathan Rosenberg. You may call me at (301) 405-5166 or (301) 405-5059 or reach me by email at jmr@math.umd.edu. I usually attend the Geometry/Topology Seminar (Mondays 3-4), the Math/Physics joint workshop Tuesday 2-3, the Representation Theory Seminar (Wednesdays 2-3), and the Algebra Seminar (Wednesdays 3-4). My office hours are Mondays and Fridays 1-2 in room 1106 (these may sometimes have to be changed, depending on when various meetings are scheduled).

Meetings:

MWF at 11 in MTH 0102.

Prerequisites:

MATH 730 and 734 or equivalent.

Text:

A Concise Course in Algebraic Topology by J. Peter May, Chicago Lectures in Math., Univ. of Chicago Press, 1999. Also useful are Hatcher's series of books on algebraic topology, which are available on the web.

Quick Summary:

This course will attempt to bridge the 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.

Schedule (will be updated as the semester progresses):

Week Topic Reading Assignment Notes
9/2-9/6 Introduction, language, foundations M, § 2.1-2.6, Ch. 5 Classes start on Wednesday.
9/9-9/13 Cofibrations and fibrations M, Ch. 6-7 See homework assignment 1, due 9/18.
9/16-9/20 Homotopy groups and exact sequences M, Ch. 8-9 Monday is Yom Kippur. See homework assignment 2, due 9/25.
9/23-9/27 Homotopy groups and exact sequences (cont'd) M, Ch. 8-9 See homework assignment 3, due 10/2. You may download a solution of problem 1 here.
9/30-10/4 CW complexes and Whitehead's Theorem M, Ch. 10-11; H-AT, § 4.1, pp. 346-360 See homework assignment 4, due 10/16.
10/7-10/11 The homotopy excision and the Hurewicz theorems M, Ch. 11 and 15; H-AT, § 4.2, pp. 360-375  
10/14-10/18 The homotopy excision and the Hurewicz theorems (cont'd) M, Ch. 11 and 15, also § 16.1-16.4; H-AT, § 4.2, pp. 360-375 See homework assignment 5, due 10/23.
10/21-10/25 Eilenberg-MacLane spaces and beginnings of obstruction theory M, § 16.4, 16.5, 18.5; H-AT, § 4.3, pp. 410-419 See homework assignment 6, due 10/30.
10/28-11/1 Postnikov systems, The Serre spectral sequence M, § 22.4; H-SS See homework assignment 7, due 11/11.
11/4-11/8 The Serre spectral sequence: more details and applications H-SS  
11/11-11/15 Serre classes and applications H-SS  
11/18-11/22 Vector bundles and classifying spaces M, § 23.1; H-VB, § 1.1-1.2  
11/25-11/29 Vector bundles and classifying spaces (cont'd)   No class November 27 or 29. Happy Thanksgiving!
12/2-12/6 Characteristic classes   See homework assignment 8, due 12/11.
12/9-12/13 Characteristic classes (cont'd)   Last week of classes
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".