TTT101: Leicester

University of Leicester.

Supported by the London Mathematical Society.

Date

24 November 2016

Speakers

Ulrich Pennig (Cardiff)
Unit spectra of K-theory via strongly self-absorbing C*-algebras

Abstract: Complex topological K-theory can be obtained from a commutative symmetric ring spectrum KU. There is a generalisation of the group of units GL_1(R) of a commutative ring R to commutative ring spectra, which is particularly interesting for K-theory, since the first group [X, BGL_1(KU)] of the associated cohomology theory classifies the twists of K-theory.

Unfortunately, the standard construction of GL_1(KU) does not directly give a geometric interpretation for this group. I will speak about an operator algebraic model for [X, BGL_1(KU)] and related groups in terms of bundles of (stabilised) strongly self-absorbing C*-algebras.

The proof that the classifying space of these bundles has the right homotopy type is based on the I-monoid model for GL_1(KU) developed by Sagave and Schlichtkrull. I will keep the material self-contained, so no prior knowledge of C*-algebras is required to follow the talk. This is joint work with Marius Dadarlat (Purdue).

Tom Coyne (Queen Mary University of London)
Singular chains on topological stacks

Abstract: We shall review the theory of topological stacks and consider a few different approaches to defining homotopy invariants. In particular, we shall consider a generalisation of the singular chains functor from topological spaces to topological stacks.

Ieke Moerdijk (Utrecht and Sheffield)
Simplicial presheaves and Quillen's theorem B

Abstract: Quillen's original theorem B can be interpreted as giving a small and explicit description of the homotopy fiber for certain types of maps between simplicial sets. We will present an extension of this result to the homotopy theories of simplicial presheaves over a site, and hope to include various applications, such as the group completion for presheaves of simplicial monoids, and the homotopy descent property for simplicial presheaves.