## **
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:
- J. Milnor,
* Morse theory *
- 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.

- R. Gompf, A. Stipsicz,
* 4-manifolds and Kirby calculus *
- D. Rolfsen,
* Knots and Links *
- N. Saveliev,
* Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant *
- 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 The second is a bit outdated and focuses on knots, but does contain some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.**

**PROBLEM SHEET** NOW UPDATED!

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