*Algebraic Topology* Fall 2017

### Instructor: Danny Calegari; Graders: Weinan Lin and Margaret Nichols

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

### Description of course:

This course is an introduction to Algebraic Topology. It is intended for
first year graduate students.

### Cancellations:

None yet.

### Notices:

None yet.

### 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 Hatcher; the notation x:y means problem y from section x.

- Homework 1, due Friday, October 6: 0:4, 0:6, 0:16, 0:20, 0:23, 1.1:6, 1.1:13, 1.1:20
- Homework 2, due Friday, October 13: 1.2:9, 1.2:11, 1.2:14, 1.2:16, 1.2:22, 1.3:10, 1.3:11
- Homework 3, due Friday, October 20: 1.3:13, 1.3:18, 1.3:19, 1.3:20, 1.3:23, 1.A:6, 1.A:13

### Notes:

None yet.

### Syllabus:

It is expected that students taking the class have taken an undergraduate algebraic
topology class before; consequently (and because time is limited and the number of
topics to cover is large) we will move through the material quickly, leaving some important
details to the homework.

The skeleton of the syllabus is the following. Some topics will be
covered very briefly.

- Fundamental group: van Kampen's theorem, covering spaces, K(G,1)'s
- Homology: simplicial, singular, cellular; Mayer-Vietoris; Axiomatic approach
- Cohomology: universal coefficients, cup product, Poincare duality

If there is time, I hope to get to higher homotopy groups, including Whitehead
and Hurewicz theorems, and some of the theory of fibrations.

### References:

The main reference is Algebraic Topology
by Allen Hatcher. Hatcher's book is very geometric and conversational, and besides includes a huge amount of material; but
his style does not appeal to everyone (especially those who like a more axiomatic approach). Some other
introductory books on algebraic topology are listed below (the first book by Bott and Tu follows a very unusual trajectory, and depends on the reader having some background in differential topology; it is not recommended for someone
without this background).