Amenable category and complexity

Capovilla, Pietro; Loeh, Clara; Moraschini, Marco

doi:10.48550/arxiv.2012.00612

Cited by 3 publications

(5 citation statements)

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“…In the absolute case, every cover is a weakly convex relative cover and hence mult Am (X, ∅) is the minimal multiplicity of amenable covers of X. For a CWcomplex X, this coincides with the minimal cardinality of amenable covers of X by not necessarily path-connected subsets [CLM20,Remark 3.13]. The latter quantity is called the amenable category cat Am (X) (Remark 4.15), a notion that is modelled on the classical LS-category [CLOT03].…”

Section: 2mentioning

confidence: 96%

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Bounded acyclicity and relative simplicial volume

Li¹,

Loeh²,

Moraschini³

2022

Preprint

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We provide new vanishing and glueing results for relative simplicial volume, following up on two current themes in bounded cohomology: The passage from amenable groups to boundedly acyclic groups and the use of equivariant topology.More precisely, we consider equivariant nerve pairs and relative classifying spaces for families of subgroups. Typically, we apply this to uniformly boundedly acyclic families of subgroups. Our methods also lead to vanishing results for ℓ 2 -Betti numbers of aspherical CW-pairs with small relative amenable category and to a relative version of a result by Dranishnikov and Rudyak concerning mapping degrees and the inheritance of freeness of fundamental groups.

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Section: 2mentioning

confidence: 96%

“…One of the classical applications of simplicial volume is an a priori estimate on mapping degrees [Gro82,Thu79,FM21]. In contrast, the exact relation between mapping degrees and monotonicity of (generalised) LS-category invariants is still wide open [Rud17,CLM20].…”

Section: 2mentioning

confidence: 99%

Bounded acyclicity and relative simplicial volume

Li¹,

Loeh²,

Moraschini³

2022

Preprint

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“…A useful approach to investigate the vanishing of simplicial volume is to consider amenable covers. This idea dates back to Gromov [51] and it was then developed further by many authors [3,21,44,47,50,60,62,76,79,93]. We use the terminology of amenable category [21,50,76]: Definition 3.4 (Amenable covers and category)…”

Section: Amenable Covers: the Closed Casementioning

confidence: 99%

“…The vanishing results for open amenable covers are usually stated in terms of assumptions on the multiplicity of the cover instead of the cardinality. These assumptions essentially are the same when working with paracompact Hausdorff spaces [21,Remark 3.13].…”

Section: Amenable Covers: the Closed Casementioning

confidence: 99%

On the simplicial volume and the Euler characteristic of (aspherical) manifolds

2022

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A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov’s question and collect strategies towards affirmative answers and strategies towards negative answers to this problem. Moreover, we put Gromov’s question into context with other open problems in low- and high-dimensional topology. A special emphasis is put on a comparative analysis of the additivity properties of the simplicial volume and the Euler characteristic for manifolds with boundary. We explain that the simplicial volume defines a symmetric monoidal functor (TQFT) on the amenable cobordism category, but not on the whole cobordism category. In addition, using known computations of simplicial volumes, we conclude that the fundamental group of the four-dimensional amenable cobordism category is not finitely generated. We also consider new variations of Gromov’s question. Specifically, we show that counterexamples exist among aspherical spaces that are only homology equivalent to oriented closed connected manifolds.

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“…In this article, we extend the class of positive examples by aspherical manifolds that admit amenable open covers of small multiplicity: If M admits an open amenable cover of multiplicity at most n, we write cat Am M ≤ n. Usually the amenable category cat Am of M is defined in terms of the cardinality of amenable covers. However, in the case of CW-complexes the two definitions are equivalent [10,Remark 3.13;13,Lemma A.4].…”

Section: Introductionmentioning

confidence: 99%