# TTT32: Manchester

Department of Mathematics, University of Manchester.

## Date

19 November 2001

## Speakers

Frank Neumann (Leicester)

Etale homotopy and moduli stacks'

Imma Galvez (Sheffield)

Elliptic genera and invariants of manifolds with boundary

Thomas Huetteman (Aberdeen)

Polytopes, homotopy colimits, and algebraic K-theory

## Abstracts

### Etale homotopy and moduli stacks

Using the machinery of etale homotopy theory a la Artin-Mazur we determine the etale homotopy type of certain moduli stacks over the rationals parametrising algebraic curves having fixed finite subgroups of its automorphism groups, which can be realised in the complex analytic case as finite subgroups of the mapping class group.

After giving an overview on Deligne-Mumford stacks and how to define its etale homotopy type, we show how we can actually determine it in this concrete algebro-geometric situation via comparison with the complex analytic case of families of Riemann surfaces with symmetries and their Teichmueller theory.

### Polytopes, homotopy colimits, and algebraic K-theory

The aim of this talk is to present a link between combinatorial topology, homotopy theory, and algebraic K-theory of spaces. My constructions are motivated by some aspects of toric geometry and might be considered as a (rather naive) attempt to "do algebraic geometry over the sphere spectrum". Given a polytope with integral vertices, there is an associated (projective) toric variety which comes equipped with a covering by spectra of monoid rings.

Thus the category of quasi-coherent sheaves is equivalent to a certain diagram category of modules over these rings. The point is that the covering and the monoids are determined by the given polytope: the "shape" of the diagrams is given by the face lattice of the polytope, the monoids make use of the actual geometry (and not just of the combinatorial structure).

Taking this observation as a starting point, I define an analogous category of "quasi-coherent homotopy sheaves of topological spaces" (a certain diagram category of equivariant spaces). The attempt to analyse its algebraic K-theory leads to the study of certain homotopy colimits which can be calculated using our knowledge of polytopes. Reading the analogy between algebra and topology backwards, one can also define homotopy sheaves in the algebraic setting.

I hope that the improved flexibility of homotopy sheaves (as compared to sheaves in the usual sense) helps to gain insight into the Quillen K-theory of toric varieties.

## Further information

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