Computing cohomology groups that classify bundles of strongly self-absorbing $C^*$-algebras

TL;DR

This paper computes cohomology groups classifying bundles of strongly self-absorbing $C^*$-algebras over finite CW-complexes, relating them to classical cohomology and $K$-theory, and explores the unit spectrum of complex $K$-theory.

Contribution

It provides explicit computations of cohomology groups for bundles of strongly self-absorbing $C^*$-algebras and compares the $C^*$-algebraic and classical $K$-theory unit spectra.

Findings

Cohomology groups are expressed in terms of ordinary cohomology and connective $K$-theory.

A uniqueness result for the unit spectrum of complex periodic topological $K$-theory is developed.

Explicit formulas for classifying bundles of strongly self-absorbing $C^*$-algebras.

Abstract

Locally trivial bundles of $C^*$-algebras with fibre $D \otimes \mathcal{K}$ for a strongly self-absorbing $C^*$-algebra $D$ over a finite CW-complex $X$ form a group $E^1_D(X)$ that is the first group of a cohomology theory $E^*_D(X)$. In this paper we compute these groups by expressing them in terms of ordinary cohomology and connective $K$-theory. To compare the $C^*$-algebraic version of $gl_1(KU)$ with its classical counterpart we also develop a uniqueness result for the unit spectrum of complex periodic topological $K$-theory.