# TTT78: Leicester

University of Leicester.

Supported by the London Mathematical Society.

## Date

18 April 2011

## Speakers

### Martin Palmer (University of Oxford)

Homological stability for oriented configuration spaces

Homological stability for oriented configuration spaces

The behaviour of the homology of unordered configuration spaces, as the number of particles n goes to infinity, is well-understood: for each degree i, the i-th homology is eventually independent of n. The new twist in this talk will be to study 'oriented' configuration spaces, in which a configuration of particles additionally possesses an ordering up to even permutations.

This is an example of a global parameter on configuration spaces; it can also be interpreted as studying the homology of unordered configuration spaces with certain twisted coefficients.

We will show that these spaces also exhibit homological stability (although interestingly only at a slower rate), while being careful to explain where the proof differs from the unordered case. We will also mention some new corollaries which are obtained for homological stability of certain sequences of groups.

### Michal Adamaszek (University of Warwick)

The symmetric join operad

The symmetric join operad

The cohomology of a space is a graded-commutative algebra with product induced by the cup-product of cochains. The latter is commutative up to homotopy via a collection of cup-i-products and higher multi-variable operations. This structure is encoded in the notion of an algebra over an E_\infty operad. We produce such an operad and its action from a geometric object.

First we study a canonical operad in the category of symmetric simplicial sets. It arises from the interaction between the product and the join of spaces and, in some sense, it coacts naturally on any simplicial set. Applying cochains to these geometric constructions gives an E_\infty operad and its natural action on the cochain complex of a space. It is closely related to the sequence operad of McClure and Smith.

Joint work with J D S Jones.

### Harry Ullman (University of Sheffield)

The equivariant stable homotopy theory of isometries

The equivariant stable homotopy theory of isometries

Non-equivariantly, a space of linear isometries admits a stable splitting. In an equivariant setting, however, this does not generally happen. Instead, one can naturally build an equivariant stable tower with interesting topological properties similar to those exhibited by the non-equivariant splitting.

We discuss this construction, while also mentioning obstructions to producing an equivariant splitting. Finally, we mention work-in-progress on retrieving a stable splitting from the tower in the special case where an equivariant splitting is possible.

## Further information

This meeting included the start of BMC programme.