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The Serre spectral sequence in bounded cohomology

Kevin Li, Marco Moraschini, George Raptis

TL;DR

The paper develops a Serre-type spectral sequence for bounded cohomology with seminormed local coefficients on simplicial sets, extending classical Serre theory beyond ordinary cohomology. It relies on Dress's formulation and a bimodal bisimplicial-complex framework to identify the $E_2$-page as $H^p_b(Y; \mathcal{H}^q_b(F; \mathcal{A}))$ under finiteness or uniform boundary conditions, and proves homotopy invariance and functoriality within the $\mathcal{L}^*$ setting. The authors then adapt the construction to $\ell^1$-homology and derive several key applications: non-isometric versions of Gromov's mapping theorem, estimates for simplicial volume in manifold bundles, and recoveries of the LHS spectral sequence for group extensions. The framework connects bounded cohomology of spaces, groups, and simplicial sets, enabling transfer of results across these contexts and providing a versatile tool for computations in geometric topology. The work is intended to facilitate new bounds and vanishing results in bounded cohomology, with potential impact on understanding fibrations, amenability phenomena, and geometric invariants such as simplicial volume.

Abstract

We construct the analogue of the Serre spectral sequence for the bounded cohomology of simplicial sets with seminormed local coefficients. As applications, we obtain a (non-isometric) generalization of Gromov's mapping theorem and some partial results on the simplicial volume of manifold bundles.

The Serre spectral sequence in bounded cohomology

TL;DR

The paper develops a Serre-type spectral sequence for bounded cohomology with seminormed local coefficients on simplicial sets, extending classical Serre theory beyond ordinary cohomology. It relies on Dress's formulation and a bimodal bisimplicial-complex framework to identify the -page as under finiteness or uniform boundary conditions, and proves homotopy invariance and functoriality within the setting. The authors then adapt the construction to -homology and derive several key applications: non-isometric versions of Gromov's mapping theorem, estimates for simplicial volume in manifold bundles, and recoveries of the LHS spectral sequence for group extensions. The framework connects bounded cohomology of spaces, groups, and simplicial sets, enabling transfer of results across these contexts and providing a versatile tool for computations in geometric topology. The work is intended to facilitate new bounds and vanishing results in bounded cohomology, with potential impact on understanding fibrations, amenability phenomena, and geometric invariants such as simplicial volume.

Abstract

We construct the analogue of the Serre spectral sequence for the bounded cohomology of simplicial sets with seminormed local coefficients. As applications, we obtain a (non-isometric) generalization of Gromov's mapping theorem and some partial results on the simplicial volume of manifold bundles.

Paper Structure

This paper contains 22 sections, 39 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Bounded cohomology of simplicial sets with local coefficients
  3. Preliminaries on simplicial sets
  4. Seminormed local coefficient systems
  5. Bounded cohomology of simplicial sets with local coefficients
  6. Bounded cohomology and bounded products
  7. Homotopy invariance of bounded cohomology
  8. Homotopy equivalences in L*
  9. Homotopy invariance of bounded cohomology
  10. Local coefficient system associated to a Kan fibration
  11. Serre spectral sequence in bounded cohomology
  12. Bisimplicial set of a map
  13. Serre spectral sequence
  14. Functoriality
  15. Serre spectral sequence in l¹-homology
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $f\colon X\to Y$ be a Kan fibration of simplicial sets and let $\mathcal{A}$ be a seminormed local coefficient system on $X$. Then there is a first-quadrant spectral sequence $(E^{\bullet,\bullet}_r)_r$ converging to $H^*_b(X;\mathcal{A})$. We have an isomorphism of seminormed $R$-modules in each of the following cases:

Theorems & Definitions (100)

  • Theorem 1.1: Theorem \ref{['thm:sss']}
  • Theorem 1.2: Theorem \ref{['thm:mapping thm class']}
  • Theorem 1.3: vbc
  • Example 2.1: Nerve of a category
  • Example 2.2: Simplex category of a simplicial set
  • Example 2.3: Moore--Postnikov tower
  • Definition 2.4: Fundamental groupoid of a simplicial set
  • Definition 2.5: Seminormed local coefficient system
  • Definition 2.6: Category of simplicial sets with local coefficients
  • Remark 2.7
  • ...and 90 more