Simplicial volume via foliated simplices and duality

TL;DR

This paper introduces a new foliated homology theory for manifolds using measure-preserving group actions, establishing a duality with bounded cohomology and providing criteria for simplicial volume vanishing.

Contribution

It develops the theory of singular foliated homology and bounded cohomology, linking them to classical invariants and extending Gromov's ideas to foliated settings.

Findings

Foliated fundamental class norm equals the simplicial volume.

Established isometric isomorphism with measurable bounded cohomology for aspherical manifolds.

Derived vanishing criteria for simplicial volume via foliated bounded cohomology.

Abstract

Let $M$ be a triangulated oriented closed connected manifold with universal cover $\widetilde{M}\to M$ and fundamental group $\Gamma=\pi_1(M)$ and consider an essentially free measure preserving action $\Gamma\curvearrowright (X,\mu)$ on a standard Borel probability space. We study the space $\Gamma\backslash(\widetilde{M}\times X)$ equipped with the measured foliation defined by Sauer and the theory of singular foliated simplices in this setting. We define its real singular foliated homology and compare it to classical singular homology. In particular, we construct a foliated fundamental class and prove that its norm coincides with the simplicial volume of $M$, formalizing ideas of Gromov. Passing to the dual chain complex, we define the singular foliated bounded cohomology. When $M$ is aspherical we establish an isometric isomorphism with the measurable bounded cohomology of the action groupoid $\Gamma\curvearrowright X$. As a consequence of a foliated duality principle, we odeduce vanishing criteria for the simplicial volume of $M$ in terms of the vanishing of the measurable bounded cohomology of the action groupoid and/or of its transverse groupoids.