A characteristic class of \(\mathrm{Homeo(X)_0}\)-bundles and an abelian extension of the homeomorphism group

Abstract

A \(\mathrm{Homeo(X)_0}\)-bundle is a fiber bundle with fiber \(X\) whose structure group reduces to the identity component \(\mathrm{Homeo(X)_0}\) of the homeomorphism group of \(X\). We construct a characteristic class of \(\mathrm{Homeo(X)_0}\)-bundles as a third cohomology class with coefficients in \(\mathbb{Z}\). We also investigate the relation between the universal characteristic class of flat fiber bundles and the gauge group extension of the homeomorphism group. Furthermore, under some assumptions, we construct and study the central \(S^1\)-extension and the corresponding group two-cocycle of \(\mathrm{Homeo(X)_0}\).