MA9M7 Topics in Geometric Topology
Term II 2022-2023

Module Description

We will take a stroll through the field of three-manifolds, heading in the general direction of the distant peaks of the homeomorphism problem and the geometrisation theorem. More concretely: we will begin the module with a focus on classic combinatoral topology of three-manifolds (including their sphere and torus decompositions) in order to understand the hypotheses of the geometrization theorem. We will then switch to a discussion of the eight Thurston geometries in order to understand the conclusion of the geometrisation theorem. We will end with an overview of some more advanced topics (Mostow rigidity, Ricci flow, and Perelman's proof) and their application to the homeomorphism problem. Throughout the module we place an emphasis on the many beautiful examples that the subject offers.

The prerequisites for the module are as follows. Point-set topology, including connectedness, compactness, continuity, homeomorphism, and manifolds. Algebraic topology, including covering spaces, the fundamental group, the classification of surfaces, homology, and Euler characteristic. Spherical, euclidean, and hyperbolic geometry in dimension two, including the classification of isometries and geometric surfaces.

Schedule

The schedule has a planned list of topics, organized by lecture. We will change the schedule as necessary, as we work through the material. Links to example sheets will be posted week-by-week.

Instructor

Class meetings

Reference materials

Useful references include the notes on three-manifolds by Andrew Casson, Cameron Gordon, Allen Hatcher, Marc Lackenby (scroll down to Michaelmas 1999), and Danny Calegari. Of course, there is also Thurston's book, in its various versions.

I will post copies of the lectures in the class schedule as they become available.

Example sheets

See the schedule for the example sheets. Hatcher's notes and Thurston's book also contain many interesting exercises.

Assessed work

There is no exam for this module. If you are taking the module for credit you must complete ten homework exercises (each worth ten marks). The due dates are flexible, but everything should be in by the end of week eleven.

Mistakes

Please tell me in person, or via email, about any errors on this website or made in class. I am especially keen to hear about mathematical errors, gaffes, or typos made in lecture or in the example sheets.