The Dixmier-Douady classes and abelian extensions of groups of homeomorphisms

Let X be a connected topological space and c∈H2(X;Z) be a non-zero cohomology class. A Homeo(X,c)-bundle is a fiber bundle with fiber X whose structure group reduces to the group Homeo(X,c) of c-preserving homeomorphisms of X. If H1(X;Z)=0, then a characteristic class for Homeo(X,c)-bundles called t...

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Bibliographic Details
Published inTopology and its applications Vol. 336; p. 108600
Main Author Maruyama, Shuhei
Format Journal Article
LanguageEnglish
Published Elsevier B.V 15.08.2023
Subjects
Dixmier-Douady class
Gauge group
Dixmier-Douady class
55R40
Gauge group
57R20
20J06
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Summary:Let X be a connected topological space and c∈H2(X;Z) be a non-zero cohomology class. A Homeo(X,c)-bundle is a fiber bundle with fiber X whose structure group reduces to the group Homeo(X,c) of c-preserving homeomorphisms of X. If H1(X;Z)=0, then a characteristic class for Homeo(X,c)-bundles called the Dixmier-Douady class is defined via the Serre spectral sequence. We show the relation between the universal Dixmier-Douady class for foliated Homeo(X,c)-bundles and the gauge group extension of Homeo(X,c). Moreover, under some assumptions, we construct a central S1-extension and a group two-cocycle on Homeo(X,c) corresponding to the Dixmier-Douady class.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2023.108600