# TTT33: Leicester

Department of Mathematics and Computer Science, University of Leicester.

## Date

1 March 2002

## Speakers

Joel Segal (Canterbury)

Invariant theory on rings of divided powers

Nick Wright (Penn State)

The coarse Baum-Connes conjecture via C_0 coarse geometry

Hansjoerg Geiges (Cologne, visiting Cambridge)

Surgery presentations of contact 3-manifolds

## Abstracts

### Invariant theory on rings of divided powers

The ring of polynomials S(V*) over a finite field has a dual, the ring of divided powers D(V). This is dual in several ways, in particular as a Hopf algebra and as the dual construction from the tensor algebra.

D(V) has been studied to some extent, notably in representation theory and as a tool in understanding the Steenrod algebra action on S(V*). It is a functor, natural in the vector space, and as such any linear group action on V extends to an action on D(V) in the same way as for polynomial rings. However D(V) is not finitely generated as an algebra, and every element is a zero divisor, making the algebra much less tractable than the polynomial case.

The polynomials invariant under the action of a finite group S(V*)^G have been much studied, notably by Hilbert and Noether, but also by modern algebraists and topologists. This talk will deal with the beginnings of the study of the invariant divided powers D(V)^G, and connections between the two.

### Surgery presentations of contact 3-manifolds

A contact structure on a 3-manifold is a totally non-integrable tangent 2-plane field (the opposite of a foliation, as it were). A construction due to Lutz and Martinet from the 1970s, based on surgery along knots {\em transverse} to a given contact structure (the standard contact structure on the 3-sphere, for instance), shows how to produce – on any given orientable 3-manifold – a contact structure in each homotopy class of 2-plane fields.

But it does not answer the question: Which contact structures can be obtained by this construction?

In this talk, I describe the Lutz-Martinet construction and an alternative form of contact surgery along knots {\em tangent} to a given contact structure, which is strong enough to produce any contact structure. These are results due to Fan Ding and the speaker.

No previous knowledge of contact geometry is assumed.

### The coarse Baum-Connes conjecture via C_0 coarse geometry

The coarse Baum-Connes conjecture asserts that a certain assembly map gives an isomorphism from the coarse K-homology of a space to the K-theory of its coarse C^*-algebra. The conjecture has applications outside of coarse geometry, for example to the Novikov conjecture.

In this talk I introduce alternative and more refined coarse structures arising from a metric, and reformulate the assembly map as a forgetful functor which forgets this refinement. I demonstrate that for spaces of finite asymptotic dimension an interpolation between these structures can be used to show that this forgetful functor is an isomorphism, giving a new proof of the conjecture for this class of spaces.

## Further information

The meeting was partially supported by a Scheme 3 grant from the London Mathematical Society, and the EU Modern Homotopy Theory RTN.