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Amenable covers and relative bounded cohomology

Pietro Capovilla

TL;DR

The paper advances the relative vanishing theory for bounded cohomology by establishing a relative Gromov vanishing result under amenable covers of small multiplicity, and by connecting this to a nerve-based description when the cover is convex. It uses Gromov's multicomplex framework and mapping-cone techniques to translate between (X,A) and corresponding multicomplex pairs, enabling a transfer of cohomological information and the construction of a nerve map Theta^n. The main contributions are: (i) a vanishing of the relative bounded cohomology comparison map $\mathrm{comp}^n$ for $n\geq \mathrm{mult}_A(\mathcal{U})$ under amenable kernel assumptions, and (ii) a nerve-based refinement providing a commutative diagram with $N(\mathcal{U})$ and $N(\mathcal{U}_A)$ when the cover is convex. These results generalize prior work by Li, Löh, and Moraschini and relate to vanishing phenomena for relative simplicial volume, clarifying the role of RC1/RC2 regularity in the amenable-cover setting.

Abstract

We establish a relative version of Gromov's Vanishing Theorem in the presence of amenable open covers with small multiplicity, extending a result of Li, Löh, and Moraschini. Our approach relies on Gromov's theory of multicomplexes.

Amenable covers and relative bounded cohomology

TL;DR

The paper advances the relative vanishing theory for bounded cohomology by establishing a relative Gromov vanishing result under amenable covers of small multiplicity, and by connecting this to a nerve-based description when the cover is convex. It uses Gromov's multicomplex framework and mapping-cone techniques to translate between (X,A) and corresponding multicomplex pairs, enabling a transfer of cohomological information and the construction of a nerve map Theta^n. The main contributions are: (i) a vanishing of the relative bounded cohomology comparison map for under amenable kernel assumptions, and (ii) a nerve-based refinement providing a commutative diagram with and when the cover is convex. These results generalize prior work by Li, Löh, and Moraschini and relate to vanishing phenomena for relative simplicial volume, clarifying the role of RC1/RC2 regularity in the amenable-cover setting.

Abstract

We establish a relative version of Gromov's Vanishing Theorem in the presence of amenable open covers with small multiplicity, extending a result of Li, Löh, and Moraschini. Our approach relies on Gromov's theory of multicomplexes.
Paper Structure (15 sections, 11 theorems, 14 equations, 1 figure)

This paper contains 15 sections, 11 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Open covers
  4. Relative bounded cohomology of pairs
  5. Multicomplexes
  6. Invariant chains
  7. The singular multicomplex
  8. The group $\Pi(X,X)$ and its action on $\mathcal{A}(X)$
  9. From pairs of spaces to pairs of multicomplexes
  10. Mapping cones
  11. Proof of Proposition \ref{['prop: relative mapping theorem']}
  12. Proof of Theorem \ref{['thm: main theorem']}
  13. Proof of Theorem \ref{['thm: relative vanishing theorem: rel mult and nerves']}
  14. Proof of (1)
  15. Proof of (2)

Key Result

Theorem 1

Let $(X,A)$ be a triangulable pair and assume that the kernel of the morphism $\pi_1(A\hookrightarrow X, x)$ is amenable for every $x \in A$. Let $\mathcal{U}$ be an amenable cover of $X$ by path-connected open subsets such that: Then the comparison map $\mathrm{comp}^n\colon H^n_b(X,A;\mathbb{R})\rightarrow H^n(X,A;\mathbb{R})$ vanishes for every $n \geq \mathrm{mult}(\mathcal{U})$.

Figures (1)

  • Figure 1: These open covers of $S$ are amenable and have multiplicity 2. The one on the left satisfies (RC1), but not (RC2), the one on the right satisfies (RC2), but not (RC1).

Theorems & Definitions (22)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4: FM23
  • ...and 12 more