Math 541: Topics in Topology Fall 2006

TTh 1-2:15 PM, HB 423

Professor Tim Cochran
Herman Brown 416
(713) 348-5265 (my office) (713) 348-4829 (math office)
Office hours: TBA and by appointment.
Textbook for first part of course: Gompf and Stipsicz, 4-Manifolds and Kirby Calculus, Grad. Studies in Math. #20, Amer. Math. Soc. 1999.

This course will provide background for numerous advanced topics in topology that are currently attracting a lot of research attention in topology of manifolds. I am open to varying topics to suit the interests of the students. There will be homework to reinforce basic examples and basic notions. Optimally students should already know what the tangent bundle of a smooth manifold is and have some experience with vector bundles. However, due to the very concrete nature of the subject, most of this can be "picked up". The broad topics will include:

1. 4-dimensional manifolds (the largest topic)

a. examples including complex algebraic surfaces
b. classification of simply-connected 4-manifolds up to homotopy type
c. comparison with classification of 2-manifolds and higher-dimensional manifolds including h-cobordism theorem and Whitney trick
d. topological classification of simply-connected 4-manifolds
e.smooth techniques and existence of multiple differentiable structures
f. other structures on smooth 4-manifolds

2. Khovanov Homology of Knots and Links

3. Noncommutative Algebraic Methods in Knot and Link Theory