Math 866-867-868: Differential Topology
This is the year long sequence in differential topology. Differential topology focuses on understanding the structure of manifolds, using techniques that are primarily geometric in nature. As such, many of the results in differential topology are easier to visualize and comprehend than their counterparts in algebraic topology.
During the year long course, we will be discussing:
First quarter (866): topics will include smooth manifolds, immersions, submersions, transversality, inverse function theorem, Sard's theorem, basic Morse theory, basic intersection theory, vector fields and the Poincare-Hopf index theorem. Other topics will be added if there is time.
Second quarter (867): topics will include real vector bundles, Grassmanians and universal bundles, Stiefel-Whitney classes, oriented bundles and Euler class, Thom isomorphism theorem, obstruction theory, complex vector bundles, Chern classes, Pontrjagin classes, applications. Other topics will be added if there is time.
Third quarter (868): topics will include external algebras and differential forms, integration on manifolds, Stokes theorem, de Rham cohomology, Poincare duality, Laplacian and harmonic forms,
basic Hodge theory. Other topics will be added if there is time.
Problem Sets (866): Basic Differential Topology
Problem Sets (867): Characteristic Classes
Problem Sets (868): Characteristic Classes & Differential forms
We won't have any required textbooks for the course, though there are a number of "recommended textbooks" that will make good supplementary reading:
J. Milnor, Topology from the differentiable viewpoint. An inexpensive, very readable introduction to the subject. Highly recommended to everyone. It provides a very gentle introduction to many of the key ideas in the field, and will help you get a feel for the sort of mathematics this course will cover.
V. Guillemin and A. Pollack, Differential topology. One of the standard textbooks on the subject. Also quite readable, and covers most of the material covered in 866, and some of the material from 868.
M. Hirsch, Differential topology. One of the other standard textbooks on the subject. This book covers most of the material covered in 866, and some of the material for 867. This book is harder to read than Guillemin-Pollack, but covers quite a bit of additional material.
M. Spivak, Calculus on manifolds. This is a small, relatively inexpensive book. It covers some of the material from 866 and from 868. It is quite readable, and can also serve as a good review of ordinary differentiation and integration, prior to moving to the manifold case.
R. Bott and L. Tu, Differential forms in algebraic topology. This is a great book, and covers a lot of the material for 867, and some of the material for 868. It also includes a wealth of additional information which we will definitely not cover (but is still good to know about).
S. Morita, Geometry of differential forms. This is a recent book which covers a lot of the material for 867, and some of the material for 868. I am not as familiar with it as with the other books in this list, but I've been told it is quite good.
J. Milnor and J. Stasheff, Characteristic classes. This fantastic book is the standard text on this subject, and covers all of the material that we will discuss in 867, and some of the material from 868. It is very well written, and very readable.
Grading will be mostly based on homework problem sets, which will be assigned roughly on a weekly basis. Each problem set will have approximately four problems, and students are expected to complete at least half the problems to keep a "satisfactory" standing in the course. In addition, there will be one take home final exam at the end of each quarter.
My office is in MW 510 (Math Tower). Office hours are MWF from 10:30-11:18, or by appointment.
If you need to contact me, the most efficient way is via e-mail at email@example.com