Math 276 - Section 1 - Topics in Topology

Instructor: Ian Agol
Lectures: TuTh 9:30-11:00am, Room 51 Evans
Course Control Number: 54758
Office: 921 Evans
Office Hours: TBA

Class will be cancelled T/Th Jan. 26, 28, because of MSRI meeting

On Thursday, February 18, Ken Bromberg will give a guest lecture on Dehn surgeries on the figure 8 knot complement.

Class will be cancelled April 20 & 22.
We will probably have some make-up classes May 4 & 6 during the reading period

Prerequisites: Algebraic Topology, Riemannian Geometry, would
be helpful to have attended Math 277 fall 2009.

Required Text: There will be notes made available over the
course of the semester.

Homework:
Due 2/2/10: Go through chapter 3 of Kirby's problem list, and determine which problems
follow as consequences of the geometrization or orbifolds theorems.

Due 2/9/10:  Classify covers with cyclic fundamental group of closed aspherical 3-manifolds

Due 2/16/10: Describe the geometric decomposition and a non-positively curved Riemannian metric with
locally Euclidean geodesic boundary on the 4 chain link complement.

Recommended Reading:
Peter Scott, The geometries of 3-manifolds

William Thurston, Three-dimensional Geometry and Topology, Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5.
Pirated preliminary draft book (12.2 MB scanned copy)

Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P. Three-dimensional Orbifolds and Cone-manifolds. With a Postface by Sadayoshi Kojima, MSJ Memoirs, 5. Mathematical Society of Japan, Tokyo, 2000. x+170 pp. ISBN 4-931469-05-1.

Boileau, Maillot, Porti, Three dimensional orbifolds and their geometric structures

Boileau, Leeb, Porti, Geometrization of 3-orbifolds

Sources on the proof of Geometrization using Ricci flow

The Geometrization conjecture and universal covers of 3-manifolds
talk at the Cornell Topology Festival, 2004

Further Reading:
Jaco's lectures on homeomorphism classification of 3-manifolds

Lackenby's Three-dimensional manifolds (Graduate Course, Michaelmas 1999)

Hatcher, Notes on basic 3-manifold topology

Thurston, The Geometry and Topology of 3-manifolds

Syllabus: This seminar will focus on topics in 3-manifold
topology which are consequences of the Geometrization Theorem and
Orbifold Theorem, which we will take for the most part as a black box.
Some topics may include:

- review of Thurston's 8 geometries and the geometrization and orbifold theorems

- classification of universal covers of closed 3-manifolds

- geometric proofs of Dehn's lemma, the loop theorem, and sphere theorem

- homotopy rigidity of aspherical 3-manifolds

- solution to the word and conjugacy problems in 3-manifold fundamental groups

- algorithmic classification of compact 3-manifolds

- homotopy equivalence classficiation, and simply homotopy equivalence => homeomorphic for closed 3-manifolds

If there is time, we may survey some other topics such as the
classification of covers of compact 3-manifolds with finitely generated fundamental group, the generalized Smale conjecture, Dehn surgery results, and volume estimates for Haken hyperbolic 3-manifolds.