# TTT81: Sheffield

The University of Sheffield.

Supported by the London Mathematical Society.

This was a special meeting aimed at postgraduates and postdocs. Many thanks to David Barnes and Pokman Cheung for organising this TTT.

## Date

4 November 2011

## Speakers

### Eugenia Cheng (Sheffield)

Introduction to operads and loop spaces

Introduction to operads and loop spaces

One of the basic ways of producing algebra from topology is the fundamental group, whose elements are homotopy classes of loops. However, if we wish to avoid quotienting out by homotopy we need a more subtle construction to deal with the fact that concatenation of loops is not associative. Operads provide a convenient way of keeping track of the resulting algebra which is only associative 'up to homotopy'.

In this talk we will introduce operads as a way of handling operations of different arities. We will explain how operads can be used to recognise when a given space 'is' a loop space, that is, the space of loops of another space.

Finally we will discuss how operads can be regarded as algebraic theories of a specific kind which does not admit the theory of groups. As loop spaces are 'groups up to homotopy' this means that operads cannot be used to define the whole theory of loop spaces.

### David Barnes (Sheffield)

Spheres and stability, equivariant spheres and equivariant stability

Spheres and stability, equivariant spheres and equivariant stability

Spheres are the building blocks of homotopy theory. If we allow 'negative spheres' we obtain the notion of stable homotopy theory. Working stably we are able to see many fascinating patterns and structures within homotopy theory. We now want to generalise this to spaces with an action of a compact Lie group.

We discuss what kinds of spheres are needed to build equivariant homotopy theory and what kinds of spheres we wish to invert to make a good notion of stable equivariant homotopy theory.

### Ana Lucia Garcia-Pulido (Warwick)

Models for string topology

Models for string topology

In this talk I will describe a general method of calculating the Hochschild cohomology for a graded commutative algebra. I will discuss analogous results regarding Hochschild homology.

### Alastair Darby (Manchester)

Quasitoric Manifolds

Quasitoric Manifolds

Quasitoric manifolds are one of the main objects of study in the emerging field of Toric Topology. They are a certain class of manifolds, with a torus action, that can be defined purely in terms of combinatorial data. It can be shown that these manifolds admit a canonical stably complex structure and constitute a sufficiently wide class of stably complex manifolds to additively generate the complex cobordism ring.

I will show how these manifolds are constructed once given the combinatorial data and how we realise its stably complex structure.