# TTT74: Manchester

School of Mathematics, Alan Turing Building, University of Manchester.

Supported by the London Mathematical Society and MIMS.

## Date

5 July 2010

## Speakers

### Alex Gonzalez (Sheffield)

Unstable Adams operations acting on p-local compact groups and fixed points

Unstable Adams operations acting on p-local compact groups and fixed points

p-local compact groups were recently introduced by C Broto, R Levi and B Oliver as algebraic models for p-completions of classifying spaces of compact Lie groups and p-compact groups. In this new setting, a definition of unstable Adams operations for p-local compact groups was provided in F Junod, along with a proof of their existence in all cases.

We then study the action of such operations on a fixed p-local compact group, and study whether the corresponding fixed points form a p-local finite group. The importance of such a result is great: first, this would provide a unifying statement and proof for results on compact Lie groups and p-compact groups. Also, this would provide a powerful tool to extend known results on p-local finite groups to the compact case. Some examples will be discussed at the end.

### Jeff Giansiracusa (Swansea)

Modular operads and diffeomorphisms of 3-dimensional handlebodies

Modular operads and diffeomorphisms of 3-dimensional handlebodies

Configurations spaces of points in the plane form an operad, and configuration spaces of framed points form a cyclic operad (the roles of inputs and outputs can be exchanged). I will describe how the modular operad generated by this cyclic operad gives a model for BDiffs of 3- dimensional handlebodies. This leads to a graph complex computing the cohomology of these BDiffs.

### Ronnie Brown (Bangor)

Some strict higher homotopy groupoids: intuitions, examples, applications, prospects

Some strict higher homotopy groupoids: intuitions, examples, applications, prospects

The aim is to show how the idea of 'algebraic inverse to subdivision' led to a family of strict higher homotopy groupoids more intuitive and powerful than the earlier relative homo- topy groups, through having structure in a full range of dimensions and also the advantages of symmetry and multiple compositions.

These structures help not only to understand traditional 1 structures of such homotopy groups, such as actions but can allow specific calculations of some such groups through calculation of richer structures, modelling the n-types. Even richer structures allow calculations of say Whitehead products and new results such as an n-adic Hurewicz theorem.

## Further information

The meeting was jointly supported by the London Mathematical Society and MIMS.