# TTT39: Leicester

Department of Mathematics and Computer Science, University of Leicester.

## Date

4 March 2003

## Speakers

Simona Paoli (Warwick)

(Co)homology of crossed modules

Javier Gutierrez (UAB Barcelona and Sheffield)

Homotopy localisations of Eilenberg-MacLane spectra

Pascal Lambrechts (Louvain-la-Neuve)

Configuration spaces from the rational homotopy viewpoint

## Abstracts

### (Co)homology of crossed modules

Crossed modules are algebraic models of 2-types. These objects have topological aspects as well as purely algebraic ones, and this gives rise to different (co)homology theories: the cohomology of the classifying space and the CCG cohomology, which is a type of cotriple (co)homology.

After recalling the construction of these theories I shall illustrate some further developments in the (co)homology of crossed modules; these are obtained by considering more general classes of coefficients for the cotriple (co)homology and by introducing a (co)homology of cat$^n$-groups.

### Homotopy localisations of Eilenberg--MacLane spectra

We prove that stable homotopical localisations preserve ring spectra and module spectra structures under suitable hypothesis. We use this fact to describe the main features of localization of $HR$-modules (ie, stable $R$-GEMs), motivated by similar results in unstable homotopy.

In particular, we compute the homological localisations of the Eilenberg--Mac\,Lane spectra $HG$ and describe all possible localisations of the integral Eilenberg--Mac\,Lane spectrum $H\mathbb{Z}$.

### Configuration spaces from the rational homotopy viewpoint

I will discuss on the configuration space of $k$ points in a closed manifold $M$, denoted by $F(M,k)$. Fulton and McPherson have constructed an algebraic model for the rational homotopy type of $F(M,k)$ when $M$ is a smooth complex projective algebraic variety.

In this talk we will discuss how this model can be generalised for other closed manifold, even though we cannot yet determine completely the rational homotopy type of $F(M,k)$. This is a joint work with Don Stanley.

## Further information

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