On three-manifolds dominated by circle bundles

Kotschick, D.; Neofytidis, Christoforos

doi:10.1007/s00209-012-1055-3

Cited by 26 publications

(37 citation statements)

References 21 publications

(59 reference statements)

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“…In [23] Wang suggested an ordering of all closed 3-manifolds. According to Wang's work and to the results of [11] we have the following: Theorem 2.1 (Wang's ordering [23,11]). Let the following classes of closed oriented 3-manifolds:…”

Section: Wang's Ordering In Dimension Threementioning

confidence: 99%

Ordering Thurston’s geometries by maps of nonzero degree

Neofytidis

2018

J. Topol. Anal.

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Section: Wang's Ordering In Dimension Threementioning

confidence: 99%

Ordering Thurston’s geometries by maps of nonzero degree

Neofytidis

2018

J. Topol. Anal.

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“…M = (# m S 2 × S 1 )#(# n i=1 S 3 /Q i ), where Q i are finite groups, then M is rationally inessential (i.e. its classifying map is trivial in degree three rational homology) and finitely covered by a connected sum #S 2 × S 1 (this covering corresponds to the kernel of the projection π 1 (M) −→ Q 1 × · · · × Q n [15]). Thus the mapping torus of every homeomorphism f : M −→ M is also rationally inessential and so it has zero simplicial volume.…”

Section: 1mentioning

confidence: 99%

The simplicial volume of mapping tori of 3-manifolds

Bucher¹,

Neofytidis²

2019

Math. Ann.

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We prove that any mapping torus of a closed 3-manifold has zero simplicial volume. When the fiber is a prime 3-manifold, classification results can be applied to show vanishing of the simplicial volume, however the case of reducible fibers is by far more subtle. We thus analyse the possible self-homeomorphisms of reducible 3-manifolds, and use this analysis to produce an explicit representative of the fundamental class of the corresponding mapping tori. To this end, we introduce a new technique for understanding self-homeomorphisms of connected sums in arbitrary dimensions on the level of classifying spaces and for computing the simplicial volume. In particular, we extend our computations to mapping tori of certain connected sums in higher dimensions. Our main result completes the picture for the vanishing of the simplicial volume of fiber bundles in dimension four. Moreover, we deduce that dimension four together with the trivial case of dimension two are the only dimensions where all mapping tori have vanishing simplicial volume. As a group theoretic consequence, we derive an alternative proof of the fact that the fundamental group G of a mapping torus of a 3-manifold M is Gromov hyperbolic if and only if M is virtually a connected sum #S 2 × S 1 and G does not contain Z 2 .

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“…The inspiration for the construction of the branched covering of Theorem 1 stems from a previous joint work with Kotschick on domination for three-manifolds by circle bundles [16]. More precisely, we proved there that for every k there is a π 1 -surjective branched double covering…”

Section: Construction Of Branched Double Coveringsmentioning

confidence: 92%

“…On the other hand, the fundamental group conditions given in [15] are not always sufficient to deduce domination by products, cf. [16,17].…”

Section: Theorem 1 For N ≥ 4 and Every K There Is A Branched Double mentioning

confidence: 99%

Branched coverings of simply connected manifolds

Neofytidis¹

2014

Topology and its Applications

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ABSTRACT. We construct branched double coverings by certain direct products of manifolds for connected sums of copies of sphere bundles over the 2-sphere. As an application we answer a question of Kotschick and Löh up to dimension five. More precisely, we show that (1) every simply connected, closed four-manifold admits a branched double covering by a product of the circle with a connected sum of copies of S 2 × S 1 , followed by a collapsing map; (2) every simply connected, closed five-manifold admits a branched double covering by a product of the circle with a connected sum of copies of S 3 × S 1 , followed by a map whose degree is determined by the torsion of the second integral homology group of the target.

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On three-manifolds dominated by circle bundles

Cited by 26 publications

References 21 publications

Ordering Thurston’s geometries by maps of nonzero degree

Ordering Thurston’s geometries by maps of nonzero degree

The simplicial volume of mapping tori of 3-manifolds

Branched coverings of simply connected manifolds

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