*The 4-Dimensional Poincare Conjecture* Fall 2018

### Instructor: Danny Calegari

### TR 11:00-12:20 Eckhart 117

### Description of course:

This course aims to give a careful complete proof of the (topological)
4-Dimensional Poincare Conjecture, after Freedman. This is one of the
crowning achievements of 20th century topology, but the details of the
argument are appreciated by distressingly few mathematicians. There are
essentially no prerequisites for the course other than the willingness to
stare for a long time at complicated pictures.

### Cancellations:

Tuesday October 9 there will be no class.

### Notices:

None yet.

### Syllabus:

We begin with an introduction to the `higher-dimensional' theory, giving
the engulfing proof of Stallings-Zeeman, and the h-cobordism proof of
Smale. We then introduce the theory of decompositions, culminating in a
proof of the sphere-to-sphere theorem. We move on to gropes, and the
combinatorial theory of Casson handles, enabling us to prove the
reimbedding theorem. Finally we put these pieces together and show how they
can be used to deduce the Poincare Conjecture.

### Notes from class:

The notes from class are being polished and turned into a monograph. The latest
draft version is available

here.

### References:

The main references are:

- M. Freedman and F. Quinn, Topology of 4-manifolds
- M. Freedman, lectures at Santa Barbara January-March 2013; archived at the MPI Bonn
- R. Kirby, The Topology of 4-Manifolds; Springer LNM 1374