The Dixmier-Douady class and an abelian extension of the homeomorphism group

TL;DR

This paper explores the Dixmier-Douady class in relation to $c$-preserving homeomorphism groups, establishing connections with universal classes, gauge group extensions, and constructing associated central extensions and cocycles.

Contribution

It introduces a new relation between the universal Dixmier-Douady class and gauge group extensions, and constructs explicit central $S^1$-extensions for $ ext{Homeo}(X,c)$.

Findings

Relation between universal Dixmier-Douady class and gauge group extension

Construction of a central $S^1$-extension of $ ext{Homeo}(X,c)$

Explicit group two-cocycle corresponding to the Dixmier-Douady class

Abstract

Let $X$ be a connected topological space and $c \in \mathrm{H}^2(X;\mathbb{Z})$ a non-zero cohomology class. A $\mathrm{Homeo}(X,c)$-bundle is a fiber bundle with fiber $X$ whose structure group reduces to the group $\mathrm{Homeo}(X,c)$ of $c$-preserving homeomorphisms of $X$. If $\mathrm{H}^1(X;\mathbb{Z}) = 0$, then a characteristic class for $\mathrm{Homeo}(X,c)$-bundles called the Dixmier-Douady class is defined via the Serre spectral sequence. We show a relation between the universal Dixmier-Douady class for foliated $\mathrm{Homeo}(X,c)$-bundles and the gauge group extension of $\mathrm{Homeo}(X,c)$. Moreover, under some assumptions, we construct a central $S^1$-extension and a group two-cocycle on $\mathrm{Homeo}(X,c)$ corresponding to the Dixmier-Douady class.