Volume and Euler classes in bounded cohomology of transformation groups

Abstract

Let $M$ be an oriented smooth manifold, and $\mathrm{Homeo}(M,\omega )$ the group of measure preserving homeomorphisms of $M$, where $\omega$ is a finite measure induced by a volume form. In this paper we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group ${\mathrm{Homeo}}_0(M,\omega )$ and $\mathrm{Homeo}(M,\omega )$ respectively, and in several cases prove their non-triviality. More precisely, we define: - Volume classes in $H_b^n({\mathrm{Homeo}}_0(M,\omega ))$ where $M$ is a hyperbolic manifold of dimension $n$. - Euler classes in $H_b^2(\mathrm{Homeo}(S,\omega ))$ where $S$ is a closed hyperbolic surface. We show that Euler classes have positive norms for any closed hyperbolic $S$ and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$-manifolds, and hence they are non-trivial.

arXiv commentNew chapter on couplings, minor inaccuracies and notation issues corrected. Accepted for publication in Glasgow Mathematical Journal

References in corpus (1)

• Bounded Cohomology Classes of Exact Forms