# TTT51: Sheffield

Department of Pure Mathematics, University of Sheffield.

This was a working meeting with the usual time for participant discussion.

## Date

21 October 2005

## Speakers

Jeff Giansiracusa

Mapping class groups of simply-connected 4-manifolds and characteristic classes

Richard Hepworth

Kreck-Stolz invariants

Marcello Felisatti (Leicester)

## Abstracts

### Mapping class groups of simply-connected 4-manifolds and characteristic classes

We study the mapping class groups of simply-connected 4-manifolds. By taking connected sums with CP^2 \# \overline{CP^2} one can define a stable mapping class group. We show that this stable group is in fact independent of the initial manifold, and it is isomorphic to the stabilised group of automorphisms of the intersection form.

As a corollary, the homotopy type of a cobordism category of simply-connected 4-manifolds is the Hermitian K-theory of the integers. We then consider the (unstable) cohomology of mapping class groups of 4-manifolds and describe progress towards proving a version of Harer's stability in dimension 4.

### Kreck-Stolz invariants

In 1988, Kreck and Stolz introduced new invariants of certain 7-manifolds and showed that the invariants completely classify the manifolds. I will give a generalisation of these results and describe how one can interpret the invariants using K-theory and secondary operations (and perhaps tmf).

## Further information

The meeting was partially supported by a Scheme 3 grant from the London Mathematical Society.