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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 structuregroup reduces to the group $\mathrm{Homeo}(X,c)$ of $c$-preservinghomeomorphisms of $X$. If $\mathrm{H}^1(X;\mathbb{Z}) = 0$, then acharacteristic class for $\mathrm{Homeo}(X,c)$-bundles called theDixmier-Douady class is defined via the Serre spectral sequence. We show arelation 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 $\mathrm{Homeo}(X,c)$ correspondingto the Dixmier-Douady class.

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