##### Department of Mathematics,

University of California San Diego

****************************

### Special Seminar

## Scott Morrison

#### Microsoft Station Q

## The $s$-invariant of the Cappell-Shaneson Spheres

##### Abstract:

The smooth 4-dimensional Poincare conjecture is something of an orphan. No significant progress has been made in a while, and no one is even really sure whether it's true or false. Some plausible counterexamples have been known for over 20 years, and I'll tell you about a particular family of these, the Cappell-Shaneson spheres, which we've recently been thinking about again. The obvious approach to a counterexample is to find an invariant which distinguishes it from the standard 4-sphere; sadly no such invariants are known. We're taking a different approach by extracting a `local' problem, involving the slice genus of certain knots and links. Rasmussen's $s$-invariant, related to the Khovanov homology of a link, gives bounds on the slice genus, and thence a potential obstruction. Unfortunately, the links are huge, and calculating the s-invariant is hard. Nevertheless we've made some progress (a potientially dangerous shortcut, a new algorithm, and a new method of extracting the s-invariant), and even have an answer in one case. (With Michael Freedman, Robert Gompf and Kevin Walker)

Host: Justin Roberts

### December 4, 2008

### 1:40 PM

### AP&M 6218

****************************