# TTT43: Manchester

Department of Mathematics, University of Manchester.

## Date

29 January 2004

## Speakers

Matthias Franz (Geneva)

The cohomology of smooth toric varieties

Dietrich Notbohm (Leicester)

On the homotopy type of Davis-Januszkiewicz spaces

Peter Symonds (UMIST)

Cohomology of profinite groups

## Abstracts

### The cohomology of smooth toric varieties

Smooth toric varieties are certain (not necessarily compact) complex manifolds with an action of a torus. They admit a combinatorial description in terms of convex-geometric data. We showed how to recover the integral cohomology of a smooth toric variety from its equivariant cohomology. This is actually a special case of a more general result relating (equivariant) cohomology to Koszul duality.

### On the homotopy type of Davis-Januszkiewicz spaces

For an arbitrary simplicial complex $K$, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of $K$. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples.

It is therefore natural to investigate the extent to which the homotopy type of a space $X$ is determined by having such a cohomology ring. We discussed the associated rational and p-adic homotopy uiniquess question separately. Finally we applied Sullivan's arithmetique square to produce global results in special families of cases.

### Cohomology of profinite groups

We considered various properties of the cohomology of profinite groups. In the discrete case an important role is played by a contractible space of finite dimension on which the group acts with finite stabilisers. We developed an algebraic analogue for profinite groups.

## Further information

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