Non-Zero Degree Maps Between 2n-Manifolds

Duan, Hai Bao; Wang, Shi Cheng

doi:10.1007/s10114-003-0307-x

Cited by 31 publications

(42 citation statements)

References 6 publications

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“…This result fails in dimension four, by a recent work of Pankka and Souto [19], where it is shown that T 4 is not a branched covering of # 3 (S 2 × S 2 ), while every integer can be realized as the degree for a map from T 4 to # 3 (S 2 × S 2 ) (by a criterion of Duan and Wang [8]; see Section 5.2).…”

Section: Theorem 4 Every Simply Connected Closed Four-manifold M Admentioning

confidence: 86%

“…Nevertheless, we note that T 2 × Σ k is a branched four-fold cover of # k (S 2 × S 2 ), by Remark 2. The existence of dominant maps from products T 2 × Σ k to every simply connected four-manifold has also been shown by Kotschick and Löh [15], using a result of Duan and Wang [8]. That result states that if X and Y are oriented, closed four-manifolds and Y is simply connected, then a degree d = 0 map f : X −→ Y exists if and only if the intersection form of Y , multiplied by d, is embedded into the intersection form of X, where the embedding is given by H * (f ) (the "only if" part is obvious).…”

Section: Branched Coverings In Dimension Fourmentioning

confidence: 91%

“…In dimension four, Kotschick and Löh [15] have previously obtained a non-constructive proof for the above statement based on a result of Duan and Wang [8]. Our proof here is independent of those earlier works and it moreover gives an explicit construction of a degree two map from a product of type S 1 × (#S 2 × S 1 ) to every simply connected four-manifold, obtained as the composition of a branched double covering with a certain degree one map; cf.…”

Section: Theorem 2 Every Connected Oriented Closed Manifold With Fmentioning

confidence: 99%

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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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Section: Theorem 4 Every Simply Connected Closed Four-manifold M Admentioning

confidence: 86%

Section: Branched Coverings In Dimension Fourmentioning

confidence: 91%

Section: Theorem 2 Every Connected Oriented Closed Manifold With Fmentioning

confidence: 99%

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Branched coverings of simply connected manifolds

Neofytidis¹

2014

Topology and its Applications

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“…This is not a problem when the dimension is three. In dimension four, Duan and Wang [12] show that, when we are working on continuous maps and topological manifolds, the simply connected 4-manifolds are ordered by their intersection forms. There are topological 4-manifolds not admitting smooth structures.…”

Section: Geometric Structures Gromov Norm and Kodaira Dimensions 21mentioning

confidence: 99%

“…However, we have f : M 1 #kCP 2 → CP 2 #5CP 2 with k ≥ 5. There is no such map for k < 5 since they are simply connected which is ordered by their intersection forms by [12].Notice that Theorem 5.1 is related to (and could be viewed as 0-dimensional generalization of) the Iitaka conjecture, which states that a fiber space f : X → Z satisfies κ h (X) ≥ κ h (Z) + κ h (F ) where F is a general fiber of f . Here, an (analytic) fiber space is a proper surjective morphism with connected fibres.…”

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confidence: 99%

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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Sets of Degrees of Maps Between Su(2)-Bundles Over the 5-Sphere

Lafont

Neofytidis

2018

Transformation Groups

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Non-Zero Degree Maps Between 2n-Manifolds

Cited by 31 publications

References 6 publications

Branched coverings of simply connected manifolds

Branched coverings of simply connected manifolds

Geometric structures, Gromov norm and Kodaira dimensions

Sets of Degrees of Maps Between Su(2)-Bundles Over the 5-Sphere

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