Closed aspherical manifolds with center

Cappell, Sylvain E.; Weinberger, Shmuel; Yan, Min

doi:10.1112/jtopol/jtt023

Cited by 8 publications

(7 citation statements)

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“…(Recall that a map of degree one is π 1 -surjective.) Aspherical manifolds whose fundamental groups have infinite cyclic center are of special interest, in particular with respect to the study of circle bundles and circle actions; see [3] and the references there. We begin with two general facts about finite coverings of circle bundles: −→ B ′ , which is the desired circle bundle.…”

Section: 2mentioning

confidence: 99%

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Fundamental groups of aspherical manifolds and maps of non-zero degree

Neofytidis

2018

Groups Geom. Dyn.

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We define a new class of irreducible groups, called groups not infinite-index presentable by products or not IIPP. We prove that certain aspherical manifolds with fundamental groups not IIPP do not admit maps of non-zero degree from direct products. This extends previous results of Kotschick and Löh, providing new classes of aspherical manifolds -beyond those non-positively curved ones which were predicted by Gromov -that do not admit maps of non-zero degree from direct products.A sample application is that an aspherical geometric 4-manifold admits a map of non-zero degree from a direct product if and only if it is a virtual product itself. This completes a characterization of the product geometries due to Hillman. Along the way we prove that for certain groups the property IIPP is a criterion for reducibility. This especially implies the vanishing of the simplicial volume of the corresponding aspherical manifolds. It is shown that aspherical manifolds with reducible fundamental groups do always admit maps of non-zero degree from direct products. FUNDAMENTAL GROUPS OF ASPHERICAL MANIFOLDS AND MAPS OF NON-ZERO DEGREE 3 manifolds using maps of non-zero degree [4,38] and the monotonicity of Kodaira dimensions with respect to the existence of maps of non-zero degree [39] will be presented in a subsequent paper [28].

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Section: 2mentioning

confidence: 99%

“…Next, we show that the properties (2) and (3) are equivalent. Obviously (2) implies (3). Assume now that π 1 (M) is reducible.…”

Section: And 23 For the Detailsmentioning

confidence: 99%

Fundamental groups of aspherical manifolds and maps of non-zero degree

Neofytidis

2018

Groups Geom. Dyn.

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“…Indeed, this is true when Γ/C(Γ) is of type F by a 3-for-2 property for Poincaré spaces for fibrations of finite complexes. But it turns out that the group Γ/C(Γ) is not even torsion-free in general: For example for M one of the manifolds constructed in [CWY13] as counterexamples to a conjecture about free S 1 -actions on aspherical manifolds with non-trivial centre, Γ/C(Γ) contains a non-trivial element of order 2, and thus cannot even admit a finite dimensional model of its classifying space. While we do not know whether the fundamental groups of these manifold are Farrell-Jones groups, we still have: 7.2.1.…”

Section: Examples and Counterexamplesmentioning

confidence: 99%

A vanishing theorem for tautological classes of aspherical manifolds

Hebestreit,

Land,

Lück

et al. 2017

Preprint

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Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend only on the underlying topological block bundle, and use this to prove the vanishing of tautological classes for many bundles with fibre an aspherical manifold.

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“…We remark that for a closed aspherical manifold, such a correlation between the existence of circle actions and the nontriviality of the center of the fundamental group is part of a conjectured rigidity of aspherical manifolds going back to work of Borel. See [9] for some recent progress and more detailed discussions.…”

Section: Irreducible S 1 -Four-manifoldsmentioning

confidence: 99%

Fixed-point free circle actions on 4–manifolds

Chen

2015

Algebr. Geom. Topol.

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Abstract. This paper is concerned with fixed-point free S 1 -actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, there are at most finitely many distinct smooth (resp. topological) 4-manifolds which support a fixed-point free smooth (resp. locally linear) S 1 -action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-point free S 1 -actions under some further conditions on the fundamental group. The connection between the classification of the S 1 -manifolds and the fundamental group is given by a certain decomposition, called fiber-sum decomposition, of the S 1 -manifolds. More concretely, each fiber-sum decomposition naturally gives rise to a Z-splitting of the fundamental group. There are two technical results in this paper which play a central role in our considerations. One states that the Z-splitting is a canonical JSJ decomposition of the fundamental group in the sense of Rips and Sela [39]. Another asserts that if the fundamental group has infinite center, then the homotopy class of principal orbits of any fixed-point free S 1 -action on the 4-manifold must be infinite, unless the 4-manifold is the mapping torus of a periodic diffeomorphism of some elliptic 3-manifold.

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Closed aspherical manifolds with center

Abstract: We show that in all dimensions greater than 7 there are closed aspherical manifolds whose fundamental groups have nontrivial center but do not possess any topological circle actions. This disproves a conjectured converse (proposed by Conner and Raymond) to a classical theorem of Borel.

Cited by 8 publications

References 46 publications

Fundamental groups of aspherical manifolds and maps of non-zero degree

Fundamental groups of aspherical manifolds and maps of non-zero degree

A vanishing theorem for tautological classes of aspherical manifolds

Fixed-point free circle actions on 4–manifolds

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