# TTT44: Sheffield

Department of Pure Mathematics, the University of Sheffield.

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

## Date

19 March 2004

## Speakers

Tom Leinster (Glasgow)

Towards the algebraic topology of self-similar spaces?

Stephen Theriault (Aberdeen)

The H-structure of low rank torsion free H-spaces

John Hunton (Leicester)

Mayer Vietoris in K-theory

## Abstracts

### Towards the algebraic topology of self-similar spaces?

Homotopy and homology work well when the spaces concerned are built up by patching together small contractible pieces. On the other hand, they are near-useless for many self-similar spaces (Julia sets of holomorphic maps on Riemann surfaces being an important example).

A different kind of local-to-global patching process turns out to be appropriate in that context. I will describe it, use it to give a precise definition of self-similar space, and explain the surprising result that every compact metrisable space is self-similar. Finally, I will indicate how one day this might give suitable algebraic invariants of self-similar spaces to substitute for homotopy and homology.

### The H-structure of low rank torsion free H-space

Start with a fixed prime p and a space X of t odd dimensional cells, where t is less than p-1. After localising at p, Cooke, Harper, and Zabrodsky constructed a finite H-space Y with the property that the mod-p homology of Y is generated as an exterior Hopf algebra by the reduced mod-p homology of X.

Cohen and Neisendorfer, and later Selick and Wu, reproduced this result with different constructions. We use Selick and Wu's approach to show that Y is homotopy associative and homotopy commutative if X is a suspension and t is less than p-2. Interesting examples include some of the mod-p Stiefel manifolds considered by Mimura, Nishida, and Toda.

### Mayer Vietoris in K-theory

Suppose Y is a space composed by gluing together some other spaces. I will attempt to relate the resulting K-theories. Complications will ensue as time allows.

## Further information

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