# TTT30: Sheffield

Department of Pure Mathematics, University of Sheffield.

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

## Date

11 July 2001

## Speakers

Andy Baker (Glasgow)

Artin-Schreier extensions and the Adams spectral sequence for elliptic cohomology

Jorge Devoto (Buenos Aires)

Level 1 equivariant elliptic cohomology

Taras Panov (Manchester)

Combinatorics, topology and homotopy theory of toric spaces'

## Abstracts

Taras Panov: 'Combinatorics, topology and homotopy theory of toric spaces' (based on joint works with Victor Buchstaber and Nigel Ray)

Given a simplicial polytope K, or a more general combinatorial object such as a fan or a simplicial complex, one may produce several different types of spaces acted on by a torus out of it. The common feature of these toric spaces is that their orbit structure is determined by the combinatorics of K.

Examples include toric varieties, (quasi)toric manifolds, coordinate subspace arrangement complements and universal "moment" manifolds and complexes. Both ordinary and equivariant cohomology of these toric spaces usually can be described combinatorially, which opens the way to topological treatment of combinatorial invariants of K.

Another interesting phenomenon appearing in the equivariant topology of toric spaces is that the corresponding Borel constructions and spaces themselves admit a regular categorical description as the colimits of diagrams over face category of K in different categories.

For instance, the Borel constructions associated to all above listed toric spaces are homotopy equivalent to the colimit of the diagram of classifying spaces of tori. The cohomology of the latter colimit space is the Stanley-Reisner face ring of K appearing in the commutative algebra of simplicial complexes.

It turns out that for some nice K (namely, when K is a flag complex) the above colimit of classifying spaces is itself the classifying space for the so-called rotation group corresponding to K, which is a continuous analogue of the right-angle Coxeter group W(K).

This implies, that in the case of flag complex colimits of the diagrams of tori and classifying spaces for tori determined by K agree (in the sense that the classifying space for the colimit group is the colimit space). If K is not a flag complex, then two colimits no longer agree, and the obstructions are determined by higher Whitehead and Samelson products.

Finally, in the case of general K the inconsistence between the diagram of groups and the diagram of classifying spaces can be remedied by replacing colimits with homotopy colimits.

## Further information

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