*Differential Topology* Winter 2016

### Instructor: Danny Calegari

### MWF 11:30-12:20 Eckhart 206

### Description of course:

The first part of this course is an introduction to Characteristic Classes.
The last 3 weeks will focus on Differential Forms. The class is intended for
first year graduate students.

### Cancellations:

None yet.

### Notices:

### Homework/Midterm/Final

There will be a midterm and a final. There will also be weekly homework.
Homework is posted to this website each Friday and due at the *start* of class the following Friday. Late homework will not be accepted.

Homework is usually taken directly from the book

**Characteristic Classes**
by Milnor; the notation x:y means problem y from section x.

- Homework 1, due Friday, January 15: Milnor 1:A, 1:B, 1:C
- Homework 2, due Friday, January 22: Milnor 2:A, 2:B, 2:C, 3:C, 3:D, 3:E; also, watch Outside In
- Homework 3, due Friday, January 29: Milnor 4:A, 4:C, 4:D, 5:B, 5:C, 5:E
- Homework 4, due Friday, February 5: Milnor 6:B, 6:C, 6:D, 7:B, 7:C
- Midterm, due Friday, February 12: here
- Homework 5, due Friday, February 19: Milnor 8:A, 9:A, 9:C, 11:C, 11:D, 12:B
- Homework 6, due Friday, February 26: Milnor 14:C, 14:D, 16:B, 16:D; also read (carefully) the proof of Theorem 15.9, and (at your leisure) Chapters 16-19
- Homework 7, due Friday, March 4: here
- Homework 8, due Friday, March 11: here
- Final, due Wednesday, March 16: here

### Notes:

Notes on Differential Forms can be downloaded here and will be updated as we go along (notes updated
3/1/2016).

### Background/Syllabus:

It is expected that students taking the class have taken the fall graduate
algebraic topology class before. Thus we will assume material such as
fundamental group, homology, cohomology, higher homotopy groups,
Eilenberg-MacLane spaces (although we will recall some results as necessary).

Some familiarity with basic smooth topology is assumed (implicit function
theorem, Sard's theorem, transversality, partitions of unity), roughly the first
few chapters of Milnor's **Topology from the Differentiable Viewpoint**.

We aim to go through most of the book **Characteristic Classes** by Milnor and Stasheff.

### References:

You might also find it helpful to watch Milnor's Hedrick lectures on Differential Topology from 1965, available from the Simons Foundation
here.