Variations on the theme of the uniform boundary condition

New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in $$\mathbb {R}_{\ge 0}$$ R ≥ 0 . In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected 4-manifold. Our group theoretic results relate stable commutator length to the $$l^1$$ l 1 -semi-norm of certain singular homology classes in degree 2. The output of these results is translated into manifold constructions using cross-products and Thom realisation.

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“…Because π 1 (∂ d ) is amenable, there exists a constant K ∈ R >0 with the following property [32,51]…”

Section: Stable Filling Normsmentioning

confidence: 99%

The spectrum of simplicial volume

Heuer

Löh

2020

Invent. math.

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“…Proof By the correspondence between local and twisted coefficients on the chain level (Proposition ), this is a direct consequence of the parametrised UBC for tori formulated in terms of twisted coefficients [, Theorem 1.3].

□

…”

Section: Simplicial Volumes Of Graph Manifoldsmentioning

confidence: 99%

“…Proof We proceed as in the case with a single boundary component [, Proposition 10.3]: In order to keep the notation simple, we view

M_{1}

and

M_{2}

as subspaces of

M

and identify

T_{1}

and

T_{2}

via

f

. By the Seifert and van Kampen theorem for fundamental groupoids, the inclusions of

M_{1}

and

M_{2}

into

M

induce an isomorphism

G ≅ π (M)

.…”

Section: Simplicial Volumes Of Graph Manifoldsmentioning

confidence: 99%

“…Theorem can be proved by induction over the graph structure; the base case of a manifold with tame

S^{1}

‐structure can be solved using methods by Fauser , the induction step requires a glueing argument similar to previous results by Fauser and Löh . The main contribution of the present paper is to give a proper formalisation of this induction argument and adapting the glueing argument to the case of multiple boundary components; this will be done in the setting of fundamental groupoids and local coefficients.…”

Section: Introductionmentioning

confidence: 95%

“…In addition, the following classes of manifolds are known to satisfy integral approximation for simplicial volume: closed surfaces of genus at least 1 , closed hyperbolic 3‐manifolds , closed aspherical manifolds with residually finite amenable fundamental group , compact manifolds with ‘non‐trivial’

S^{1}

‐action , as well as certain glueings along tori . In contrast, approximation fails uniformly for higher dimensional hyperbolic manifolds and it fails for closed manifolds with non‐abelian free fundamental group [, Remark 3.9].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Integral approximation of simplicial volume of graph manifolds

Fauser

Friedl

Loeh

2019

Bull. London Math. Soc.

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Graph manifolds are manifolds that decompose along tori into pieces with a tame S 1 -structure. In this paper, we prove that the simplicial volume of graph manifolds (which is known to be zero) can be approximated by integral simplicial volumes of their finite coverings. This gives a uniform proof of the vanishing of rank gradients, Betti number gradients and torsion homology gradients for graph manifolds. Theorem 1.2. Let M be an oriented closed connected graph manifold (in the sense of Definition 2.7) with residually finite fundamental group. Then M = M ∞ Z = 0. More generally, let (Γ j ) j∈N be a descending chain of finite index subgroups of π 1 (M ) with trivial intersection and let (M j ) j∈N be the corresponding tower of finite coverings. Then M = lim j→∞ M j Z [π 1 (M ) : Γ j ] = 0.

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