# TTT111: Sheffield

The University of Sheffield.

Supported by the London Mathematical Society.

## Date

11 July 2019

## Speakers

### Neil Strickland (Sheffield)

Iterated chromatic localisation

Iterated chromatic localisation

### Wajid Mannan (QMUL)

Flattening Nancy's toy

Flattening Nancy's toy

Abstract: A fundamental question in low dimensional topology is if it is possible to have an n-dimensional homotopy type (of finite CW-complexes), which has no cohomology in dimension n or above.

A leading candidate over the last decade, colloquially referred to as 'Nancy's toy', is a certain 3-complex with no cohomology above dimension 2. Yet it was believed that strange number theoretic properties of its homotopy would prevent a homotopy equivalence with a 2-complex. I will discuss my recent proof (joint with T Popiel) that this is not the case.

### Luis Barbosa Torres (Leicester)

Equivariant cohomology of differentiable stacks and spectral sequences

Equivariant cohomology of differentiable stacks and spectral sequences

Abstract: We start with an action of a Lie group on a differentiable stack and consider the quotient stack associated to this action. Consequently, we construct an atlas that makes these quotient stacks differentiable stacks. Using the nerve of the associated Lie groupoid of that stack gives us the homotopy type for this quotient stack and the Borel model for equivariant cohomology.

We construct various spectral sequences as computational tools for the equivariant cohomology generalising the classical Bott type spectral sequence for Lie group actions on smooth manifolds. Finally, in the case of a compact Lie group we derive a Cartan model for the equivariant cohomology of a differentiable G-stack and discuss some of its properties.

### Ingrid Membrillo Solis (Southampton)

The homotopy types of gauge groups over lens spaces

The homotopy types of gauge groups over lens spaces

Abstract: Given a principal G-bundle over a space X, the gauge group is the group of its bundle automorphisms. The homotopy theory of gauge groups has received considerable attention due to their connections to geometry and mathematical physics. In this talk I will give an introduction to the homotopy classification problem of gauge groups and present some results when X is a lens space and G=U(n). This is joint work with Stephen Theriault.