Ordering Thurston’s geometries by maps of nonzero degree

Neofytidis, Christoforos

doi:10.1142/s1793525318500280

Cited by 15 publications

(18 citation statements)

References 20 publications

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“…Theorem 4.7 implies there is no non-zero degree map from M to N such that κ g (M ) < 2 and κ g (N ) = 2. Recently, [42] shows a similar result as Theorem 3.11 for closed geometric 4-manifold, i.e. κ g is monotone with respect to the existence of maps of non-zero degree.…”

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confidence: 59%

Geometric structures, Gromov norm and Kodaira dimensions

Zhang

2017

Advances in Mathematics

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A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP URL' above for details on accessing the published version and note that access may require a subscription.For more information, please contact the WRAP Team at: wrap@warwick.ac.uk GEOMETRIC STRUCTURES, GROMOV NORM AND KODAIRA DIMENSIONS WEIYI ZHANGAbstract. We define the Kodaira dimension for 3-dimensional manifolds through Thurston's eight geometries, along with a classification in terms of this Kodaira dimension. We show this is compatible with other existing Kodaira dimensions and the partial order defined by non-zero degree maps. IntroductionComplex Kodaira dimension κ h (M, J) provides a very successful classification scheme for complex manifolds. This notion is generalized by several authors (c.f. [31,39,40,26,27]) to symplectic manifolds, especially of dimension two and four. In these two dimensions, this symplectic Kodaira dimension is independent of the choice of symplectic structures [31]. In other words, it is a smooth invariant of the manifold which is thus denoted by κ s (M ). In dimension four, the smaller the symplectic Kodaira dimension, the more we know. Symplectic 4-manifolds with κ s = −∞ are diffeomorphic to rational or ruled surfaces [36]. When κ s = 0, all known examples are K3 surface, Enrique surface and T 2 bundles over T 2 . Moreover, it is shown in [31] that a symplectic manifold with κ s = 0 has the same homological invariants as one of the manifolds listed above. When κ s = 1 or 2, no classification is possible since symplectic manifolds in both categories could admit arbitrary finitely presented group as their fundamental group [19].In [9], the authors prove that complex and symplectic Kodaira dimensions are compatible with each other. More precisely, when a 4-manifold M admits at the same time both complex and symplectic structures (but the structures are not necessarily compatible with each other), then κ s (M ) = κ h (M, J). In [35], a general framework of "additivity of Kodaira dimension" is provided to further understand the compatibility of various Kodaira dimensions in possibly different dimensions. In particular, it is shown that the Kodaira dimensions are additive for fiber bundles, Lefschetz fibrations and coverings.Higher dimensional generalizations of Kodaira dimension, e.g. symplectic Kodaira dimension in dimensions six or higher, are less understood except for a proposed definition in [33]. Like complex Kodaira dimension, it will no longer be a smooth invariant. Hence, the study of this notion in higher dimensions will be associated to the study of deformation classes of symplectic structures and symplectic birational geometry.As suggested by the additivity framework, dimension three should also attach certain counterpart of Kodaira dimension. In this paper, we give a definition of Kodaira dimension κ t (M ) in dimension three through Thurston's eight ...

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confidence: 59%

Geometric structures, Gromov norm and Kodaira dimensions

Zhang

2017

Advances in Mathematics

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“…As a second ingredient for the proof of Theorem 6.7 we recall the notion of Kodaira dimension of 3-manifolds introduced by Zhang [65]. We follow here the notation by Neofytidis [53], which is slightly different from Zhang's original one. To this end we first divide Thurston's three-dimensional eight geometries into four classes assigning to each of them a value:…”

Section: Degree-one Maps and Categorical Invariantsmentioning

confidence: 99%

Amenable category and complexity

Capovilla¹,

Loeh²,

Moraschini³

2020

Preprint

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“…Nevertheless, every self-map of the circle of degree greater than one is homotopic to a non-trivial covering. The same holds for every self-map of degree greater than one of nilpotent manifolds [Be] and of certain solvable mapping tori of homeomorphisms of the n-dimensional torus [Wa1,Ne3]. In addition, every non-zero degree self-map of a closed 3-manifold M is either a homotopy equivalence or homotopic to a covering map, unless the fundamental group of each prime summand of M is finite or cyclic [Wa2].…”

Section: 3mentioning

confidence: 99%

“…The fundamental groups of our mapping tori E h are Hopfian being residually finite by Mal'cev's theorem [Ma]. More interestingly, all known examples of self-maps of aspherical manifolds of non-zero degree are either homotopic to a non-trivial covering or homotopic to a homeomorphism when the degree is ±1, as predicted by the Borel conjecture [BL,Be,BHM,FJ,Gr1,Gr2,Mi1,Mi2,Ne2,Ne3,Se1,Se2,Wa,Wa2]. It is therefore natural to ask whether this is always true, strengthening Hopf's problem for the class of aspherical manifolds [Ne2, Problem 1.2] (see Question 8.9).…”

Section: Introductionmentioning

confidence: 99%

Endomorphisms of mapping tori

Neofytidis

2020

Preprint

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Ordering Thurston’s geometries by maps of nonzero degree

Abstract: We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree.

Cited by 15 publications

References 20 publications

Geometric structures, Gromov norm and Kodaira dimensions

Geometric structures, Gromov norm and Kodaira dimensions

Amenable category and complexity

Endomorphisms of mapping tori

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