Algebraic Topology 215A, Class Information Spring 2008

Prof. Joel Hass. Office: 2222 MSB. Telephone: 752-1082.

email: hass (at symbol) math.ucdavis.edu

web page: http://math.ucdavis.edu/~hass

Class Location: 140 PhyGeo, MWF 2:10-3:00 PM

Office hours (2222 MSB): MW 3-4

Textbook: A. Hatcher, "Algebraic Topology", Cambridge University Press 2002.
You can buy this reasonably priced text. It is also available for free at Hatcher's website:
Hatcher's Book Page

We will cover Chapters 0,1 and part of Chapter 2 in Fall Quarter.

Teaching Assistant

We have a Teaching Assistant this quarter:
Emi Arima, earima (at) math, MSB 2123

Emi will lead a weekly discussion section on Monday at 12-1 in 2112 MSB.
Emi's office hours: Tuesday 2-3 PM and Fridays 10-11 AM, MSB 2123.

Exams and grading:

Your grade will be based equally on homework exercises, a final exam and a project. The project can be either
1. A lecture to the class on a topic agreed upon with the instructor.
2. A web page giving a detailed exposition on a topic agreed upon with the instructor. Something like what might be found on wikipedia.
3. A written paper giving an exposition of a topic agreed upon with the instructor

Some possible topics:

A. Choose an open problem from Kirby's list of open problems in topology. This 380 page problem list has been very influential, and is freely available at Kirby's web site. Once you choose a problem you like, you can
(a) Use mathscinet or scholar.google or the arxives to determine its current status.
(b) Describe the problem, why it is interesting, and how it might be solved.
Bonus points will be awarded if you can solve one of these problems.

B. Explain one of the following concepts:
The Kaufmann polynomial
The Alexander polynomial
The Thurston norm
Reidemeister moves
Morse function
Orientation and its relation to 1-sidedness
Smooth, topological and piecewise linear manifolds
Regular homotopies
Lens spaces
Heegaard splittings of 3-manifolds
Incompressible surfaces
Dehn's Lemma

Pick a topic in knot theory or 3-manifold theory and explain it. You can find topics in the following books.
Colin Adams, "The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots"
Erica Flapan, "When topology meets chemistry: A topological look at molecular chirality"
J. Hempel, "3-manifolds," Princeton U. Press.
W. B. Raymond Lickorish, "An Introduction to Knot Theory," Graduate Texts in Mathematics, Springer, 1997.
Dale Rolfsen, "Knots and Links", 1976.

Some recommended books on algebraic topology and related topics

Homework

HW will be listed here. HW will be due weekly on Wednesday, either by putting it in Emi Arima's mailbox by 5PM or by giving it to her in person in her office hours. Some subset of the problems will be graded. The text has a wealth of good problems, some very challenging.

You are allowed and encouraged to work with others on your homework. However you must write up your own solutions. Problems marked with an asterisk are challenging, and are optional.

HW 1 Chapter 0, p. 18: 1, 2, 3, 5, 6a*,b*, 8, 10, 16*

HW 2 Due 10/15 Chapter 0, p. 18: 17. Chapter 1, p. 38: 1, 2, 3, 5, 8

HW 3 Due 10/22 Chapter 1, p. 38: 10, 12, 13, 15, 16, 17

HW 4 Due 10/29 Chapter 1, p. 52: 2,3, 6, 7, 9, 10

HW 5 Due 11/5 Chapter 1, p. 52: 4, 5, 21. P. 79: 1,3.

HW 6 Due 11/12
1. Compute a presentation for the fundamental group of the figure eight knot. A picture can be found at the Knot Plot web site. Optional: Show that this group is not cyclic*.
2. The three edges of a triangle can be identified to a single edge to give two different complexes. If the edges are all oriented the same way, then one gets a 2-complex called the "core of a (3,1) lens space". If two are oriented one way and the third is oriented the other way, the resulting two complex is called a "dunce cap". Compute the fundamental group of each of these 2-complexes.
Chapter 1, p. 79: 4, 5,6*, 9, 10, 11.

HW 7 Due 11/19 Chapter 1, p. 79: 12, 13.

HW 8 Due 11/26 Chapter 1, p. 79: 17, 18*, 20, 25.