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

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.

arXiv comment17 pages