TTT83: Nottingham

University of Nottingham.

Supported by the London Mathematical Society.

Note the non-standard vertex! Many thanks to Mark Grant for organising TTT83.

Date

23 March 2012

Speakers

Hellen Colman (Wright College, Chicago)
A notion of LS-category internal to the category of orbifolds

Classically, an orbifold is defined as a topological space equipped with an orbifold structure given by an equivalence class of orbifold atlases. From a modern point of view, these atlases and equivalence classes are described in terms of topological groupoids and Morita equivalences.

We show that there is a Quillen model structure on the category of orbifolds considered as topological groupoids, and discuss the abstract notion of LS-category derived from this model. This is a new numerical invariant for topological groupoids which generalises the Lusternik-Schnirelmann category of topological spaces.

Michael Weiss (Aberdeen)
Smooth maps to the plane and Pontryagin classes

An obvious-looking conjecture about the rational cohomology of BTOP(n) (where TOP(n) is the group of homeomorphisms from Euclidean n-space to itself) can be reformulated as a conjecture about certain spaces of smooth regular (=nonsingular) maps to the plane.

In order to get somewhere with the reformulated conjecture, we embed spaces of smooth regular maps in spaces of smooth maps with some well-understood singularities. This follows the concordance theory and parameterised Morse theory tradition. The new aspect is that we have maps to the plane, not to the real line; and a new symmetry group O(2) instead of O(1). Joint work with Rui Reis.

Dirk Schuetz (Durham)
Configuration spaces of linkages in high dimensions

A closed linkage in d-dimensional Euclidean space consists of n segments, each of a given length, consecutively joined to each other so that the resulting broken line begins and ends at the origin. We consider the configuration space of all such closed linkages up to rotations, in particular its dependence on the given lengths of the segments.

For d=2 and 3 the (co)homology is well understood, and can be used to distinguish these spaces. For d>3, only very few results are known. We use equivariant Morse theory to study the topology of these spaces, and in particular show how the Poincare polynomials for odd d can be obtained.