Simplicial volume of products and fiber bundles

Bucher-Karlsson, Michelle

doi:10.1090/conm/501/09841

Cited by 6 publications

(8 citation statements)

References 5 publications

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“…Thus in both cases we have seen that the number of terms in 2T (0) can, up to taking boundaries, be reduced by at least a factor of 8 12 = 2 3 . Suppose now the points φ(γ 0 )x, .…”

Section: Claim 44 For Every I ∈ {1 K} There Exist Chainsmentioning

confidence: 70%

“…As observed at the end of Sect. 3, the results of the first author show that E ≥ 6χ(E) for any aspherical surface bundle E over a surface [7,8]. Now if we restrict our attention to surface bundles over surfaces coming from the ramified covering construction explained in the previous subsection, we can enhance this inequality.…”

Section: Simplicial Volume Of Such Bundlesmentioning

confidence: 96%

“…The class e has a quite explicit representative, which can be described as follows (see also [8,Section 3]). The holonomy morphism of the bundle F → E π B gives rise to the following diagram:…”

Section: The Euler Classmentioning

confidence: 99%

“…Kotschick also obtained the stronger inequality 3|σ (E)| < χ(E) in some special cases [18]. The first author's work on simplicial volume of surface bundles [8] produced an inequality between simplicial volume and Euler characteristic of aspherical surface bundles:…”

Section: Introductionmentioning

confidence: 99%

“…[8],Corollary 1.3 and [7], Corollary 3) Let F → E B be an oriented surface bundle over a surface, with closed oriented base and fibre. Then E ≥ F × B .…”

mentioning

confidence: 99%

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Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic

Bucher

Campagnolo

2020

Geom Dedicata

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“…Thus in both cases we have seen that the number of terms in 2T (0) can, up to taking boundaries, be reduced by at least a factor of 8 12 = 2 3 . Suppose now the points φ(γ 0 )x, .…”

Section: Claim 44 For Every I ∈ {1 K} There Exist Chainsmentioning

confidence: 70%

Section: Simplicial Volume Of Such Bundlesmentioning

confidence: 96%

Section: The Euler Classmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[8],Corollary 1.3 and [7], Corollary 3) Let F → E B be an oriented surface bundle over a surface, with closed oriented base and fibre. Then E ≥ F × B .…”

mentioning

confidence: 99%

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Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic

Bucher

Campagnolo

2020

Geom Dedicata

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The spectrum of simplicial volume

Heuer

Löh

2020

Invent. math.

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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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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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Simplicial volume of products and fiber bundles

Cited by 6 publications

References 5 publications

Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic

Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic

The spectrum of simplicial volume

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

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