Algebraic Topology Fall 2013

Instructor: Danny Calegari

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:

Student Disability Services is seeking to hire a paid note-taker for this class. If you are interested in this position, please send an email to Student Disability Services (notetake@uchicago.edu). In the subject line type: note-taker applicant and include the name of the class for which you wish to serve as a note-taker (MATH 31700-01).

In your email message type the following: My name is (include your full, formal first and last name), I am interested in serving as a note-taker for (MATH 31700-01). My phone number is XXX-XXX-XXXX.

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 11: Hatcher 0:4, 0:6, 0:12, 0:16, 0:21, 1.1:6, 1.1:13, 1.1:19
• Homework 2, due Friday, October 18: Hatcher 1.2:9, 1.2:22, 1.3:1, 1.3:6, 1.3:10, 1.3:13, 1.3:31
• Homework 3, due Friday, October 25: Hatcher 2.1:3, 2.1:8, 2.1:12, 2.1:16, 2.1:18, 2.1:20
• Homework 4, due Friday, November 1: Hatcher 2.1:15, 2.1:17, 2.1:22, 2.1:25, 2.1:29, 2.1.31
• Midterm, due Friday, November 8: here
• Homework 5, due Friday, November 15: Hatcher 3.1:1, 3.1:3, 3.1:5, 3.1:8, 3.1:13, 3.2:1
• Homework 6, due Friday, November 22: Hatcher 3.2:2, 3.2:4, 3.2:8, 3.2:12, 3.2:14, 3.2:15, 3.2:18
• Homework 7, due Friday, December 6: Hatcher 3.3:3, 3.3:7, 3.3:15, 3.3:30, 4.1:4, 4.1:11, 4.1:15, 4.1:17
• Final, due Friday, December 13: here

Notes:

Introductory notes on fiber bundles are available here

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).

• Bott and Tu, Differential Forms in Algebraic Topology
• Greenberg and Harper, Algebraic Topology A First Course
• May, A Concise Course in Algebraic Topology
• Spanier, Algebraic Topology