Piecewise straightening and Lipschitz simplicial volume

Strzałkowski, Karol

doi:10.1142/s1793525317500133

Cited by 2 publications

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References 9 publications

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“…The Lipschitz simplicial volume equals the classical simplicial volume for compact manifolds and is more rigid invariant than locally finite simplicial volume for non-compact manifolds-it is invariant under proper Lipschitz homotopy equivalences. On the other hand, the Lipschitz simplicial volume allows us to generalise most of the theorems concerning the simplicial volume to the non-compact case, including product inequality and proportionality principle [12,14,4].…”

Section: Introductionmentioning

confidence: 99%

Lipschitz Simplicial Volume of Connected Sums

Strzałkowski

2017

Preprint

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Section: Introductionmentioning

confidence: 99%

Lipschitz Simplicial Volume of Connected Sums

Strzałkowski

2017

Preprint

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

Frigerio¹,

Moraschini²

2023

Memoirs of the AMS

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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 ) \mathcal {K}(X) associated to a topological space X X , and we prove that the geometric realization of K ( X ) \mathcal {K}(X) is homotopy equivalent to X X for every CW complex X 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 ) \mathcal {K}(X) is complete for every CW complex X 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. 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 ℓ 1 \ell ^1 -invisibility of closed manifolds in terms of amenable covers. As an application, we give the first detailed proof of the vanishing of the simplicial volume of the product of three open manifolds.

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Piecewise straightening and Lipschitz simplicial volume

Cited by 2 publications

References 9 publications

Lipschitz Simplicial Volume of Connected Sums

Lipschitz Simplicial Volume of Connected Sums

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

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