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Simplicial volume via foliated simplices and duality

Filippo Sarti

TL;DR

This work formalizes Gromov’s foliated approach to simplicial volume by coupling a triangulated manifold $M$ with an essentially free p.m.p. action $Γ o (X,μ)$ to form the foliated space $Γackslash( ilde{M} imes X)$. It defines real singular foliated homology and a foliated fundamental class whose norm matches the classical simplicial volume $||M||$, and proves an isometric isomorphism between foliated bounded cohomology and the measurable bounded cohomology of the action groupoid $Γ times X$ when $M$ is aspherical. A duality principle yields vanishing criteria for $||M||$ via the vanishing of $ extup{H}_{ ext{mb}}^n(Γ times X;\mathbb{R})$ or of transverse groupoids, and a completion framework relates foliated cohomology to ordinary bounded cohomology of the group, enabling computation through non-equivariant chain complexes. Together, these results integrate dynamics, foliation theory, and bounded cohomology to provide a rigorous, dynamical realization of Gromov–Connes-type dualities for simplicial volume with broad implications for rigidity and vanishing phenomena.

Abstract

Let $M$ be a triangulated oriented closed connected manifold with universal cover $\widetilde{M}\to M$ and fundamental group $Γ=π_1(M)$ and consider an essentially free measure preserving action $Γ\curvearrowright (X,μ)$ on a standard Borel probability space. We study the space $Γ\backslash(\widetilde{M}\times X)$ 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 $Γ\curvearrowright X$. 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.

Simplicial volume via foliated simplices and duality

TL;DR

This work formalizes Gromov’s foliated approach to simplicial volume by coupling a triangulated manifold with an essentially free p.m.p. action to form the foliated space . It defines real singular foliated homology and a foliated fundamental class whose norm matches the classical simplicial volume , and proves an isometric isomorphism between foliated bounded cohomology and the measurable bounded cohomology of the action groupoid when is aspherical. A duality principle yields vanishing criteria for via the vanishing of or of transverse groupoids, and a completion framework relates foliated cohomology to ordinary bounded cohomology of the group, enabling computation through non-equivariant chain complexes. Together, these results integrate dynamics, foliation theory, and bounded cohomology to provide a rigorous, dynamical realization of Gromov–Connes-type dualities for simplicial volume with broad implications for rigidity and vanishing phenomena.

Abstract

Let be a triangulated oriented closed connected manifold with universal cover and fundamental group and consider an essentially free measure preserving action on a standard Borel probability space. We study the space 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 , formalizing ideas of Gromov. Passing to the dual chain complex, we define the singular foliated bounded cohomology. When is aspherical we establish an isometric isomorphism with the measurable bounded cohomology of the action groupoid . As a consequence of a foliated duality principle, we odeduce vanishing criteria for the simplicial volume of in terms of the vanishing of the measurable bounded cohomology of the action groupoid and/or of its transverse groupoids.

Paper Structure

This paper contains 16 sections, 18 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Acknowledgements.
  4. Preliminaries
  5. Bounded cohomology of groups, spaces and actions.
  6. Homological algebra.
  7. Fundamental classes and simplicial volume
  8. $\mathcal{R}$-simplicial complexes, $\mathcal{R}$-spaces, foliated simplices and homology
  9. $X$-spaces and $\mathcal{R}$-simplicial complexes
  10. $\mathcal{R}$-spaces
  11. Foliated singular simplices and homology
  12. Functoriality
  13. Foliated homology and simplicial volume
  14. A dual point of view
  15. Singular foliated bounded cohomology.
  16. ...and 1 more sections

Key Result

Theorem 1

Let $M$ be a triangulated oriented closed connected $n$-manifold. Let $p: \widetilde{M}\to M$ be its universal cover and consider an essentially free measure preserving action $\Gamma=\pi_1(M)\curvearrowright (X,\mu)$ on a standard Borel probability space. Then

Theorems & Definitions (50)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Gromov, Ivanov
  • Definition 2.4
  • Lemma 2.5
  • ...and 40 more