# TTT95: Manchester

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

Supported by the London Mathematical Society and MIMS.

## Date

23 March 2015

## Speakers

### Jelena Grbic (Southampton)

Homotopy Rigidity of the Functor ΣΩ

Homotopy Rigidity of the Functor ΣΩ

The main problem of this talk is the study of the homotopy rigidity of the functor ΣΩ. Our solution to this problem depends heavily on new decompositions of looped co-H-spaces.

I shall start by recalling some classical homotopy theoretical decomposition type results. Thereafter, I shall state new achievements and discuss how new functorial decompositions of looped co-H-spaces arise from an algebraic analysis of functorial coalgebra decompositions of tensor algebras. This is a joint work with Jie Wu.

### Ilia Pirashvili (Leicester)

The fundamental groupoid as a terminal costack

The fundamental groupoid as a terminal costack

The notion of a stack, which is the 2-mathematical analogue of a sheaf, has been studied for some time. In this talk, we will introduce the notion of a costack, which is essentially what we get by reversing the arrows. We then show that this rather neglected object has a very nice property.

For a topological space X, the final object of the 2-category of costacks over X is the assignment U → Π1(X). The same also holds for the etale fundamental groupoid. This implies that we essentially get a purely categorical description of the fundamental groupoid. At the end, we will also mention some generalisations of this result.

### Hendrik Suess (Manchester)

Lower dimensional torus actions

Lower dimensional torus actions

Toric manifolds are studied from at least three viewpoints, namely that of algebraic geometry, symplectic geometry and topology. These manifolds are completely described by a convex polytope (the moment polytope).

Recently, in all three fields attempts were made to generalise this description by combinatorial data and the corresponding results for toric manifolds to the case of lower dimensional torus actions. I will describe the algebraic setting and sketch the relations to what has been done in the other fields.