Gromov's theory of multicomplexes with applications to bounded cohomology and simplicial volume

TL;DR

This paper develops the theory of multicomplexes, generalizes Gromov's bounded cohomology, and applies these concepts to prove key theorems about simplicial volume and bounded cohomology of manifolds.

Contribution

It establishes the foundational theory of multicomplexes, proves Gromov's theorems with complete proofs, and extends results to non-compact spaces and products of manifolds.

Findings

Homotopy equivalence of singular multicomplexes to CW complexes

Proof that bounded cohomology depends only on the fundamental group

Vanishing criteria for simplicial volume of open manifolds

Abstract

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in 1982. In order to study its main properties, Gromov himself initiated the dual theory of bounded cohomology, that developed into an active and independent research field. Gromov's theory of bounded cohomology was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets. In the first part of this paper we lay the foundation of the theory of multicomplexes. We construct the singular multicomplex K(X) associated to a topological space X, and we prove that K(X) is homotopy equivalent to $X$ for every CW complex X. Following Gromov, we introduce the notion of completeness, which translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes. In the second part we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. We provide complete proofs of Gromov's Mapping Theorem (which implies that the bounded cohomology of a space only depends on its fundamental group) and of Gromov's Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. The third part is devoted to the study of locally finite chains on non-compact spaces. We expand some ideas of Gromov to provide complete proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the l^1-invisibility of closed manifolds in terms of amenable covers. As an application, we give the first complete proof of the vanishing of the simplicial volume of the product of three open manifolds.