Ma 194c, Topology and Geometry of 3-manifolds

Spring 2005

Pictures by the Geometry Center and Rich Schwartz respectively.

• Time and Place: 10-11:25am Tuesdays and Thursdays, in Sloan 253b (the math conference room, you have to go through the main math office to get to it).
• Instructor: Nathan Dunfield
  • E-mail:
  • Office: Sloan 258 Office Phone: 4339
  • Office Hours: TBA and by appointment. Or just drop by – even if I can't talk to you then, we'll set up a time to meet.
• Web Page: www.its.caltech.edu/~dunfield/classes/2005/194/
• Historical Note: This class was formerly known as "Thespian Group Theory" but is under new management.

Course Description

This is an introductory course in the topology and geometry of 3-manifolds. The goal of 3-dimensional topology is to classify all compact 3-manifolds. I will begin with the topological foundations of the theory, and then move on to the geometry of 3-manifolds. Thurston's Geometrization Conjecture says that any compact 3-manifold can be decomposed into pieces which admit geometric structures. In 2003, Grisha Perelman announced a proof of Geometrization using the Ricci flow. If correct, this would be a big step forward as the Geometrization Conjecture provides a "structure theorem" which allows powerful insights into even purely topological questions about 3-manifolds. A good part of the course will be devoted to understanding this conjecture. I will also include a discussion of Perelman's approach, but likely only at a cursory and superficial level. The class will conclude with the discussion of a recent development in the theory of 3-manifolds. Possibly this will more about Perelman's work, but more probably this will be about Heegaard Floer Homology and the work of Ozsvath, Szabó, and Rasmussen. Or about something else, I make no promises in this regard.

Here is a short (4 page) introduction/outline to the course.

Texts:

Unfortunately, there is no good text for this course. I will be using the following articles and notes, which you can download from the links below.

• Hatcher, Allen. Notes on Basic 3-Manifold Topology.
• Bonahon, Francis. Geometric Structures on 3-manifolds.
• Scott, G. Peter. The geometries of 3-manifolds. Bull. London Math. Soc. 15 (1983), no. 5, 401--487.

The following two books may also be useful. Thurston's book is great but doesn't go far enough for this course. You definitely want Thurston's book if you are unfamiliar with hyperbolic geometry. Rolfsen's book takes a nice topological point of view, but for the most part the topics covered there are not central to this course.

• Thurston, William P. Three-dimensional geometry and topology. Princeton Mathematical Series, 35. Princeton University Press, 1997. $60 (list).
• Rolfsen, Dale. Knots and Links, AMS, 2003. $55 (list).

Grading:

Your course grade will be determined by a final paper. This will be a 6-8 page paper written on a topic in related to the content of this class. The topic will be chosen in consultation with me in the 4th or 5th week of class, and the final version will be due on Friday, June 3. In addition, an outline of your paper will be due at the beginning of class on Thursday, May 12, and a partial draft (2/3rds complete) will be due on Thursday, May 26. Detailed suggestions for topics will be given later.

Note: Unlike many 19x series classes, letter grades will be given, rather than the course being graded pass/fail. The intermediate stages of the paper will count towards your final grade.

Prerequisites:

This is an advanced graduate course, but at the same time not a huge amount of specific background is needed. It should be accessible to first year graduate students and advanced undergraduates.

Specifically you need to be familiar with:

• Geometric Topology: Smooth manifolds and maps, transversality, vector bundles, regular (tubular) neighborhoods. Classification of compact surfaces. Riemannian metrics and basics of Riemannian geometry. E.g. Ma 157a or Ma 109bc.
• Algebraic Topology: Fundamental groups and covering spaces, including Van Kampen's theorem. A little about higher homotopy groups, mostly just pi_2. Basic homology and cohomology. Poincare duality (which is easy to visualize in dimension 3). Eilenberg-MacLane spaces (aka K(pi,1)'s). E.g. Ma 151ab. If you have only taken Ma 109a you will probably be ok with a certain about of extra work.

Also, I will not spend much time in class on the basics of hyperbolic geometry. Those unfamiliar with hyperbolic geometry will probably need to read Chapter 2 of Thurston's book when I start talking about the geometry of 3-manifolds (about the 4th week of class).