Symmetric symplectic homotopy <i>K</i>
3 surfaces

Chen, Weimin; Kwasik, Sławomir

doi:10.1112/jtopol/jtr006

Cited by 6 publications

(4 citation statements)

References 24 publications

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“…Other family of examples can be constructed using surgeries, like infinite family of homotopy K3 surfaces, for example knot surgered symplectic homotopy K3 surfaces E(2) K , where K is any fibered knot. See [CK11] and references therein for an overview and current results in the subject.…”

Section: Some Examplesmentioning

confidence: 99%

Free immersions and panelled web 4-manifolds

Kalafat

2018

Int. J. Math.

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We show that if a compact, oriented 4-manifold admits a coassociative( * φ 0 )free immersion into R 7 then its Euler characteristic χ M and signature τ M vanish. Moreover, in the spin case the Gauss map is contractible, so that the immersed manifold is parallelizable. The proof makes use of homotopy theory in particular obstruction theory. As a further application we prove a nonexistence result for some infinite families of 4-manifolds that can not be addressed previously. We give concrete examples of parallelizable 4-manifolds with complicated non-simply-connected topology. * φ 0 = dx 1234 − dx 12−34 ∧ dx 67 − dx 13−42 ∧ dx 75 − dx 14−23 ∧ dx 56 . This is actually an example of a calibration. The subgroup of GL(7, R) that leaves * φ 0 invariant is the compact 14-dimensional Lie group G 2 . If U is the 4-plane in R 7 with last three coordinates vanishing, then * φ 0 | U = dx 1234 which is equal to vol V

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Section: Some Examplesmentioning

confidence: 99%

Free immersions and panelled web 4-manifolds

Kalafat

2018

Int. J. Math.

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“…• size of finite symmetry ( [23]) to order smooth/symplectic structures on a topological 4-manifold. The moral is that a smooth structure on a topological 4-manifold is considered the 'standard one' if it has the smallest minimal genus function, largest geometric automorphism group, or largest finite symmetry among all smooth structures.…”

Section: 31mentioning

confidence: 99%

Kodaira Dimension in Low Dimensional Topology

2015

Preprint

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“…A related question which we do not answer in this paper is whether there exists some finite subgroup G ⊂ Diff + (T 2 × S 2 ) which does not admit effective symplectic actions on (T 2 × S 2 , ω) for any choice of ω (this question is closely related to the results in [5,6,7]).…”

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confidence: 93%

“…In this paper we study effective symplectic finite group actions or, equivalently, finite subgroups of symplectomorphism groups. Despite the extraordinary development of symplectic geometry in the last three decades, the interactions between finite transformation groups and symplectic geometry seems to be so far a mostly unexplored terrain (with the remarkable exceptions of [5,6,7]).…”

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confidence: 99%

Finite groups acting symplectically on $T^2\times S^2$

Riera¹

2015

Preprint

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For any symplectic form ω on T 2 × S 2 we construct infinitely many nonisomorphic finite groups which admit effective smooth actions on T 2 × S 2 that are trivial in cohomology but which do not admit any effective symplectic action on (T 2 × S 2 , ω). We also prove that for any ω there is another symplectic form ω ′ on T 2 × S 2 and a finite group acting symplectically and effectively on (T 2 × S 2 , ω ′ ) which does not admit any effective symplectic action on (T 2 × S 2 , ω).A basic ingredient in our arguments is the study of the Jordan property of the symplectomorphism groups of T 2 × S 2 . A group G is Jordan if there exists a constant C such that any finite subgroup Γ of G contains an abelian subgroup whose index in Γ is at most C. Csikós, Pyber and Szabó proved recently that the diffeomorphism group of T 2 × S 2 is not Jordan. We prove that, in contrast, for any symplectic form ω on T 2 × S 2 the group of symplectomorphisms Symp(T 2 × S 2 , ω) is Jordan. We also give upper and lower bounds for the optimal value of the constant C in Jordan's property for Symp(T 2 ×S 2 , ω) depending on the cohomology class represented by ω. Our bounds are sharp for a large class of symplectic forms on T 2 × S 2 .

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Symmetric symplectic homotopy K 3 surfaces

Cited by 6 publications

References 24 publications

Free immersions and panelled web 4-manifolds

Free immersions and panelled web 4-manifolds

Kodaira Dimension in Low Dimensional Topology

Finite groups acting symplectically on $T^2\times S^2$

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