MAT 621 Introduction to the topology of 3- and 4-manifolds, Spring 2010.

We started with a discussion of basic Morse theory and handle decompositions. In low dimensions, handle decompositions will provide a concrete way to represent a manifold. In higher dimensions, the handles are even easier to control; "untangling" them via the Whitney trick leads to a proof of Smale's h-cobordism theorem and the Poincaré conjecture in dimensions greater than 4. The references for Morse theory are:
  1. J. Milnor, Morse theory
  2. J. Milnor, Lectures on h-cobordism theorem
Morse theory has the basic theorem (about Morse functions producing handle decompositions) in the beginning of Chapter 1, and then moves towards deep geometric results. Lectures on h-cobordism contain, apart from the proof of Smale's theorems, detailed definitions and careful proofs of of all technical lemmas on Morse functions and such.

The next part of the plan is the 3- (and 4-)manifold basics: Heegaard splittings, surgeries on knots, relation between surgeries in 3-manifolds and 4-dimensional handles, Rokhlin's thm: every 3-manifold bounds a 4-manifold, a little Kirby calculus, a brief discussion of intersection forms of 4-manifolds.

  1. R. Gompf, A. Stipsicz, 4-manifolds and Kirby calculus
  2. D. Rolfsen, Knots and Links
  3. N. Saveliev, Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant
  4. V. Prasolov, A. Sossinsky, Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology
The first book contains everything we will ever cover in class, and a lot more, in a rather encyclopaedic fashion. It is not very easy to read as the first introduction, though, and does not contain very detailed proofs. The second is a bit outdated and focuses on knots, but does contain some 3-manifold theory. The third and fourth books aim, respectively, at introducing the Casson invariant and quantum invariants, but contain some basic low-dim introductions in the first chapters. (Saveliev has more 4-manifold theory, Prasolov-Sossinsky more knots.)

Further references will be posted as the course progresses.

I'm away for two weeks, from Mar 15 until the spring break. Please try to do the following questions and meet during class time on 3/16 and 3/18 to discuss solutions.


On 3/23 and 3/25, Oleg Viro has kindly agreed to give two lectures.