*Accepted Paper*

**Inserted:** 2 jan 2020

**Last Updated:** 2 jan 2020

**Journal:** Memoirs of the American Mathematical Society

**Year:** 2019

**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.