Math 527 - Homotopy Theory
Spring 2013, Section F1
Course outline
Homotopy theory is a branch of topology that studies spaces up to continuous deformation. For example, by decreasing the radius of a disc, one can shrink it down to a point. Thus a disc is homotopy equivalent to a point, although they are different spaces. By loosening the relation among spaces, homotopy theory studies objects that are more algebraic in nature, and hence more amenable to computations. In fact, most functors introduced in algebraic topology are homotopy functors.
Math 527 builds upon an introduction to algebraic topology (e.g. Math 525 and Math 526), where algebraic invariants such as the fundamental group and singular homology and cohomology were introduced. This course will cover in more detail constructions that are homotopically well behaved. We will study some properties of the homotopy category of spaces, and discuss some computational tools.
Prerequisites
- MATH 525 and MATH 526 or equivalent.
- MATH 501 is recommended but not required.
- Basic category theory. Consult this primer for an introduction.
List of possible topics
- Homotopy groups
- Fibrations, fiber sequences
- Cofibrations, cofiber sequences
- Whitehead theorem
- Hurewicz theorem
- Eilenberg-MacLane spaces, classifying spaces
- Postnikov towers
- Obstruction theory
- Freudenthal suspension theorem, stable homotopy groups
- Blakers-Massey theorem
Time permitting:
- Serre spectral sequence
- Brown representability theorem
- Spectra
- Homotopy limits and colimits
- Simplicial sets
- Model categories
Course information
| Instructor |
Martin Frankland |
| Email |
franklan(at)illinois.edu |
| Office |
247A Illini Hall |
| Office hours |
Monday 4-5 and Tuesday 3-4 |
| Lectures |
MWF 2:00-2:50 PM in 345 Altgeld Hall |
| Course website |
http://www.math.uiuc.edu/~franklan/Math527.html |
| Textbook |
There are two required textbooks, both available online for free.
Other suggested references.
- Boris Botvinnik, Lecture Notes on Algebraic Topology. http://darkwing.uoregon.edu/~botvinn/course.pdf
- James F. Davis and Paul Kirk, Lecture Notes in Algebraic Topology. http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf
- J. Peter May and Kate Ponto, More Concise Algebraic Topology.
- Robert E. Mosher and Martin C. Tangora, Cohomology Operations and Applications in Homotopy Theory.
- Jeffrey Strom, Modern Classical Homotopy Theory.
- Robert Switzer, Algebraic Topology: Homology and Homotopy.
- Tammo tom Dieck, Algebraic Topology.
- George W. Whitehead, Elements of Homotopy Theory.
|
| Grading |
The course grade is based entirely on homework. There will be no exam.
|
| Important dates |
Mon January 14: First lecture.
Mon January 21: MLK Day. No lecture.
March 18-22: Spring break. No lectures.
Fri April 12: Drop date. (March 8 is the deadline to drop via Student Self-Service.)
Wed May 1: Last lecture. |
| Homework |
Homework assigned at each lecture will be posted on the daily log. It will be due in class, usually on Wednesday of the following week. |
| Grades |
Registered students can view their grades here. You will need your NetID and password. |
Course policies
Homework grading
- In each assignment, selected problems will be graded.
- The two lowest homework scores will be dropped.
Collaborative work
You are encouraged to work on the homework in groups, although each student should submit a separate set of solutions. Please indicate which students you collaborated with, if any.
Late homework
- Late homework will not be accepted, because solutions are posted shortly after the deadline.
- If you will be absent when homework is due, you must turn in your homework in advance. Note that I will check my mailbox (250 Altgeld) right after collecting homework in class.
- In case of documented illness or emergency, homework may be dropped.
General principle
Feel free to email me or come to office hours for any administrative issues.
Campus safety regulations
Please look at this handout about safety regulations.
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