# Algebraic Topology Fall 2016

### Description of course:

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

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### 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 7: Hatcher 0:4, 0.6, 0:16, 0:23, 1.1:6, 1.1:16, 1.1:20
• Homework 2, due Friday, October 14: Hatcher 1.2:9, 1.2:11, 1.2:16, 1.2:22, 1.3:10, 1.3:13, 1.3:23
• Homework 3, due Friday, October 21: Hatcher 1.A:6, 1.A:7, 1.A:13, 2.1:1, 2.1:3, 2.1:7, 2.1:8
• Homework 4, due Friday, October 28: Hatcher 2.1:15, 2.1:17, 2.1:22, 2.1:23, 2.1:27, 2.1:29
• Midterm, due Friday, November 4: here
• Homework 5, due Friday, November 11: Hatcher 2.2:2, 2.2:8, 2.2:16, 2.2:25, 2.2:28, 2.2:30, 2.2:35 (The Mayer-Vietoris sequence is described in section 2.2, pp.149-150)
• Homework 6, due Friday, November 18: Hatcher 3.1:1, 3.1:5, 3.1:13, 3.2:2, 3.2:3, 3.2:6, 3.2:14
• Homework 7, due Wednesday, November 23: Hatcher 3.2:15, 3.2:18, 3.3:3, 3.3:7, 3.3:12, 3.3:25
• Homework 8, due Friday, December 2: Hatcher 4.1:4, 4.1:11, 4.1:15, 4.1:17, 4.2:1, 4.2:14, 4.2:19
• Final, due Friday, December 9: here

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### 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).