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Volume and Euler classes in bounded cohomology of transformation groups

Michael Brandenbursky, Michał Marcinkowski

TL;DR

The paper constructs volume and Euler classes in the bounded cohomology of infinite-dimensional transformation groups preserving a finite measure and proves their non-triviality in several hyperbolic settings. It develops two central transfer maps, $\Gamma_b$ and its mapping-class extension $\Gamma_b^{\mathcal{M}}$, to pull bounded-cohomology classes from fundamental groups or mapping-class groups into $\operatorname{Homeo}(M,\omega)$-cohomology. By restricting these classes to free subgroups and employing l^1-homology pairings and geometric finger-pushing isotopies, the authors show $Vol_M$ (and $Vol_M^{gp}$) have positive norm in dimensions $2$ and certain dimension $3$ cases; similarly, the bounded Euler class has positive norm on suitable subgroups for hyperbolic surfaces. These results extend bounded-cohomology techniques to infinite-dimensional transformation groups and yield new invariants for hyperbolic manifolds and mapping-class dynamics, with potential implications for understanding the cohomology of diffeomorphism and homeomorphism groups.

Abstract

Let $M$ be an oriented smooth manifold, and $\operatorname{Homeo}(M,ω)$ the group of measure preserving homeomorphisms of $M$, where $ω$ 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 $\operatorname{Homeo}_0(M,ω)$ and $\operatorname{Homeo}(M,ω)$ respectively, and in several cases prove their non-triviality. More precisely, we define: - Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0(M,ω))$ where $M$ is a hyperbolic manifold of dimension $n$. - Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}(S,ω))$ 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.

Volume and Euler classes in bounded cohomology of transformation groups

TL;DR

The paper constructs volume and Euler classes in the bounded cohomology of infinite-dimensional transformation groups preserving a finite measure and proves their non-triviality in several hyperbolic settings. It develops two central transfer maps, and its mapping-class extension , to pull bounded-cohomology classes from fundamental groups or mapping-class groups into -cohomology. By restricting these classes to free subgroups and employing l^1-homology pairings and geometric finger-pushing isotopies, the authors show (and ) have positive norm in dimensions and certain dimension cases; similarly, the bounded Euler class has positive norm on suitable subgroups for hyperbolic surfaces. These results extend bounded-cohomology techniques to infinite-dimensional transformation groups and yield new invariants for hyperbolic manifolds and mapping-class dynamics, with potential implications for understanding the cohomology of diffeomorphism and homeomorphism groups.

Abstract

Let be an oriented smooth manifold, and the group of measure preserving homeomorphisms of , where 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 and respectively, and in several cases prove their non-triviality. More precisely, we define: - Volume classes in where is a hyperbolic manifold of dimension . - Euler classes in where is a closed hyperbolic surface. We show that Euler classes have positive norms for any closed hyperbolic and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic -manifolds, and hence they are non-trivial.
Paper Structure (16 sections, 10 theorems, 46 equations)

This paper contains 16 sections, 10 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounded cohomology.
  4. The bounded volume class.
  5. Euler class in the mapping class group
  6. Definitions of $\Gamma_{\space b}$ and $\Gamma_{\space b}^{ \mathcal{M}}$
  7. Description of $\Gamma_{\space b}$
  8. $\Gamma_b$ via coupling
  9. Description of $\Gamma_b^{ \mathcal{M}}$.
  10. $\Gamma_{\space b}^{ \mathcal{M}}$ via couplings
  11. Restriction to a free subgroup
  12. Volume class in dimension 2
  13. Volume class in dimension 3
  14. Euler class
  15. Proof of the theorem
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $M$ be an oriented manifold of dimension $n$ such that it is either: Suppose a measure $\omega$ is induced by a volume form on $M$ and $\omega$ is finite. Then the class $\Gamma_{\space b}(Vol_M) \in \operatorname{H}_b^n(\operatorname{Homeo}_0(M,\omega))$ has positive norm and hence is non-trivial.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 3.1
  • Example 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 4.1
  • ...and 13 more