MTH 961

This is a copy of the original webpage for this course, which I taught at MSU in 2017. All of the contact information below is out of date.

Math 961: Algebraic Topology II

Spring 2017

Instructor:	Kristen Hendricks
Office:	D320 Wells Hall
Office Hours:	T Th 3-4
E-Mail:	hendricks at math .msu .edu

A printable copy of the syllabus is here.

Location and Time

T Th 10:20-11:40 in A222 Wells.

Prerequisites

Math 960 or equivalent. (Equivalent in this case means having read and understood Chapters 1-3 and 4.1-2 of Algebraic Topology by Allen Hatcher, not including the special topics, and with the exception of the Freudenthal Suspension Theorem.)

Topics

This is a third course in algebraic topology. We will cover connections between homotopy theory and cohomology, cohomology theories, stable homotopy theory and cobordism, basic obstruction theory, K-theory and characteristic classes, and spectral sequences, possibly not in precisely that order.

Textbooks

Main textbooks for this course include:

A. Hatcher, Algebraic Topology.
A. Hatcher, Vector Bundles and K-Theory.
J. Milnor and J.Stasheff, Characteristic Classes.
A. Hatcher, Spectral Sequences.

Other texts that may be of interest include:

R. Bott and L. Tu, Differential Forms in Algebraic Topology. (Link should work on MSU system.)
J. Milnor, Topology from the Differentiable Viewpoint.
J. Davis and P. Kirk, Lecture Notes on Algebraic Topology.

Please let me know if you are having difficulty getting access to a copy of any of these references.

Homework

There will be weekly suggested exercises. These will not be collected, but you are encouraged to come talk to me about them in office hours. You will get noticeably more out of the course if you complete them promptly.

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); are there any others?

How can one distinguish between smooth manifolds that are homeomorphic but not diffeomorphic?

Suggested Homework