Gromov's Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial. Volume / Roberto Frigerio and Marco Moraschini.

Format:

Book

Author/Creator:

Frigerio, Roberto, author.

Moraschini, Marco, author.

Series:

Memoirs of the American Mathematical Society ; Volume 283.

Memoirs of the American Mathematical Society Series ; Volume 283

Language:

English

Subjects (All):

Cohomology operations.

Complexes, Semisimplicial.

Homotopy theory.

Simplexes (Mathematics).

Topological spaces.

Physical Description:

1 online resource (166 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume and bounded cohomology. In order to study the main properties of simplicial volume, Gromov himself initiated the dual theory of bounded cohomology, which then developed into a very active and independent research field. Gromov's theory of bounded cohomology of topological spaces was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets. 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 K(X) associated to a topological space X, and we prove that the geometric realization of K(X) is homotopy equivalent to X for every CW complex X. Following Gromov, we introduce the notion of completeness, which, roughly speaking, translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes, and we show that K(X) is complete for every CW complex X. 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"-- Provided by publisher.

Contents:

Multicomplexes

The singular multicomplex

The homotopy theory of complete multicomplexes

Bounded cohomology of multicomplexes

The mapping theorem

The vanishing theorem

Finiteness and vanishing theorems

Diffusion of chains

Admissible submulticomplexes of k(x)

Diffusion of locally finite chains

Some results on the simplicial volume of open manifolds.

Notes:

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Frigerio, Roberto Gromov's Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume

ISBN:

9781470474034