This is a copy of the original website for this course, which I taught at UCLA in 2015. All of the contact information below is out of date.

Math 227A: Algebraic Topology II

Spring 2015

Instructor: Kristen Hendricks
Office: 6617D Math Sciences Building
Office Hours: Immediately after class, or by appointment
E-Mail: hendricks at math .ucla .edu

A printable copy of the syllabus is here.

Location and Time

MWF 10-10:50 in Geology 6704.

Prerequisites

Math 225C or equivalent

Topics

This is a second course in algebraic topology. In the first part of the course, we will cover homotopy theory; in the second half, we will do basic characteristic classes. Because it has been a year since the corresponding first course, the speed with which we cover homotopy theory will be partly determined by student feedback.

Textbooks

A. Hatcher, Algebraic Topology (available for free here) and J. Milnor and J. Stasheff, Characteristic Classes. You may also find it helpful to consult an unfinished book of A. Hatcher, Vector Bundles and K-Theory (here), and certain portions of R. Bott and L. Tu, Differential Forms in Algebraic Topology.

Method of Evaluation

For registered students there will be four problem sets to be handed in, assigned during Weeks 2, 4, 6, and 8. There will also be weekly suggested exercises, which will not be graded, but should probably not be ignored. There will not be any exams.

Motivation

Here are a few motivating questions which can be either answered or helpfully rephrased using the techniques of this course:

Given a closed manifold M, a framed submanifold N is an embedded submanifold together with a smoothly varying basis for the normal bundle at each point. What can one say about the set of such submanifolds, up to framed cobordism in M? How does this change if the framing condition is removed?

Given a smooth manifold M of dimension m, what are the smallest integers n and k such that M may be immersed into n-dimensional Euclidean space and embedded into k-dimensional Euclidean space? (From Whitney’s theorems, n is at most 2m-1 and k is at most 2m; when can we achieve interesting lower bounds?)

For what values of n is there a bilinear multiplication on R^n without zero divisors? You have already seen the values n=1 (the real numbers), n=2 (the complex numbers), n=4 (the quaternions) and n=8 (the octonions); in fact, this is a complete list.

Homework

Week 1 Suggested Exercises.

Problem Set 1 and Week 2 Suggested Exercises.

Week 3 Suggested Exercises.

Problem Set 2 and Week 4 Suggested Exercises.

Week 5 Suggested Exercises.

Problem Set 3 and Week 6 Suggested Exercises.

Week 7 Suggested Exercises.

Problem Set 4 and Week 8 Suggested Exercises.

Week 9 Suggested Exercises.