eBook

Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume

Moraschini, Marco, Frigerio, Roberto

2023

Overview

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper The first aim of this paper is to lay the foundation of the theory of multicomplexes. After setting the main definitions, we construct the singular multicomplex In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which implies in particular 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 and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide detailed 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

• Publisher: American Mathematical Society
• Subject: Cohomology operations / Complexes, Semisimplicial / Homotopy theory / Simplexes (Mathematics) / Topological spaces
• ISBN: 1470459914, 9781470459918