Math 277: Topology and Geometry of 3-manifolds (Fall 2000)

• Instructor: Nathan Dunfield
• Time and Place: MWF 11:00am in Sever 308. (New location.)
• Homepage: http://www.math.harvard.edu/~nathand/math277/
• E-mail: nathand@math.harvard.edu.
• Office: Science Center 334. Office Phone: 5-5349.
• Office Hours: Monday: 2:30-3:30, Thursday 3-4 and by appointment.

Outline:

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. If true, this would be a big step toward classifying 3-manifolds. A good part of the course will be devoted to understanding this conjecture. Thurston proved his conjecture in the special case of Haken manifolds, and I will give an easy proof of "weak-geometrization" in this case. The last part of the course will be devoted to applications of Thurston's Geometrization Theorem for Haken manifolds. This theorem has been used to settle many old, purely topological, questions. One such application is the proof of the Smith Conjecture: Let f be a finite order diffeomorphism of the 3-sphere which has a fixed point. Then the fixed point set is the an unknotted circle and f is conjugate to a rotation in O(4).

A detailed 4-page introduction and outline: DVI, PDF.

Homework:

Here are some informal homework problems:

• Weeks 1-4 (Topology of 3-manifolds) DVI, PDF.

• Weeks 4- (Geometry of 3-manifold) DVI, PDF.

Course Notes:

A photocopy of my handwritten lecture notes are availible in the class binder in the Birkhoff Library.

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. I have also put a binder with copies of these papers in the Birkhoff Library. (including a loose photocopy of Scott's paper for easy photocopying).

• 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.
• Aitchison, I.R. and Rubinstein J.H. Localising Dehn's lemma and the loop theorem.
• Bestvina, M.; Feighn, M. A combination theorem for negatively curved groups. J. Differential Geom. 35 (1992), no. 1, 85--101.
• Shalen, Peter. Representations of 3-manifold groups.

The following two books may also be useful. The Coop should have copies of Thurston's book. Thurston's book is great but doesn't go nearly far enough for this course.

• Thurston, William P. Three-dimensional geometry and topology. Princeton Mathematical Series, 35. Princeton University Press, 1997. $45.
• Rolfsen, Dale. Knots and Links, Publish or Perish, 1990. $40.

Grading:

Post-qual math graduate students are exempt from grades. Undergraduates and pre-qual graduate students will be required to write an 8-10 page paper about some topic in 3-manifolds not covered in this course. This topic will be chosen by the student in consultation with me around the 7th week of class. The paper will be due on Friday, January 14th.

Prerequisites:

This is an advanced graduate course, but not a huge deal of background is needed. It should be accessible to first year graduate students.

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.
• 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).

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 4th week of class).

Those looking for a lower level course covering similar topics should consider Math 138.

Other notes:

No class will be held on Friday, November 3rd. An optional replacement class is available that same Friday at Columbia university in NYC, where there will be an afternoon mini-conference on 3-dimensional topology and geometry. See here for more. This precedes the #959th AMS conference that weekend, which has special sessions in Topology of 3-Manifolds, Symbolic Computation and Kleinian Groups, and Combinatorial Group Theory.

Up to Nathan's homepage.