# TTT42: Leicester

Department of Mathematics and Computer Science, University of Leicester.

## Date

1 December 2003

## Speakers

Ken Edwards LT3: Jelena Grbic (Aberdeen)

Homotopoy associative, homotopy commutative universal spaces of 2-cell complexes

Bennett LT5: David Chataur (Angers)

A bordism approach to string topology

Bennett LT5: David Chataur (Angers)

The homology of free loop spaces as a homological conformal field theory

## Abstracts

### Homotopy associative, homotopy commutative universal spaces of 2-cell complexes

For any connected space X the James construction shows that $\Omega\Sigma X$ is universal in the category of homotopy associative H-spaces in the sense that any map from X to a homotopy associative H-space Y factors through a uniquely determined H-map $F:\Omega\Sigma X \longrightarrow Y$. We investigated the universal spaces of two-cell complexes in the category of homotopy associative, homotopy commutative $H$-spaces. The universal spaces we obtain generalise a result of Cohen, Moore, and Neisendorfer concerning odd dimensional odd-primary Moore spaces.

### A bordism approach to string topology

David explained how to use geometric homology theory in order to construct the BV-structure of Chas and Sullivan on the homology of free loop spaces. All these constructions used transversality in the context of infinite dimensional manifolds.

### The homology of free loop spaces as a homological conformal field theory

One of the most exciting conjecture in string topology is the fact the homology of free loop spaces should be a homological conformal field theory, this algebraic structure is very rich. We explained that a Prop, namely the homology of the Sullivan's chord diagrams, acts on the homology of free loop spaces.

The Prop of Sullivan's chord diagrams is very close to moduli spaces of curves and this result unifies under the same algebraic structure: The BV structure due to Chas and Sullivan and the Frobenius structure of Cohen and Godin.

## Further information

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