Fundamental classes not representable by products

Kotschick, D.; Löh, Clara

doi:10.1112/jlms/jdn089

Cited by 34 publications

(108 citation statements)

References 35 publications

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“…While this is not logically necessary for the proofs of our main results, we find it convenient, following [14], to use this concept as an organizing principle. In Section 3, respectively Section 4, we then prove Theorems 1 and 2 for rationally essential, respectively inessential, three-manifolds.…”

Section: Theorem 2 a Closed Oriented Connected Three-manifold N Ismentioning

confidence: 99%

“…This property was introduced in [14] and further studied in [15] because, according to [14], it is a property that the fundamental groups of rationally essential manifolds dominated by products must have. Proof.…”

Section: Three-manifold Groups Presentable By Productsmentioning

confidence: 99%

“…The obstructions for domination by products found in [14] are applicable to rationally essential manifolds in the sense of the following definition going back to Gromov [7]: Definition 1. A closed, oriented, connected n-manifold N is called rationally essential if For three-manifolds, this definition can be interpreted in terms of the Kneser-Milnor prime decomposition [16].…”

Section: Rational Essentialness For Three-manifoldsmentioning

confidence: 99%

“…This is motivated by the work of Löh and the first author in [14,15], where strong restrictions were found for certain manifolds with large universal coverings to be dominated by products. In fact, the results of those papers show that three-manifolds dominated by products cannot have hyperbolic or Sol 3 -geometry, and must often be prime.…”

Section: Introductionmentioning

confidence: 99%

“…The study of non-zero degree maps between closed, oriented manifolds has become very active over the last few decades [2,7,14]. The existence of a non-zero degree map, M −→ N, defines a transitive relation on the set of homotopy types of closed, oriented manifolds.…”

Section: Introductionmentioning

confidence: 99%

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On three-manifolds dominated by circle bundles

Kotschick

Neofytidis

2012

Math. Z.

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We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of S 2 × S 1 . This characterization can also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group. We also determine which three-manifolds are dominated by non-trivial circle bundles, and which three-manifold groups are presentable by products.

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Section: Theorem 2 a Closed Oriented Connected Three-manifold N Ismentioning

confidence: 99%

Section: Three-manifold Groups Presentable By Productsmentioning

confidence: 99%

Section: Rational Essentialness For Three-manifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On three-manifolds dominated by circle bundles

Kotschick

Neofytidis

2012

Math. Z.

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Geometric Structures in Topology, Geometry, Global Analysis and Dynamics

Neofytidis

2012

In the Tradition of Thurston III

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A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

Moraschini

Savini

2020

Transformation Groups

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Following the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\ℍn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign.As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles.In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

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Fundamental classes not representable by products

Cited by 34 publications

References 35 publications

On three-manifolds dominated by circle bundles

On three-manifolds dominated by circle bundles

Geometric Structures in Topology, Geometry, Global Analysis and Dynamics

A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

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