Contact 3-Manifolds and Symplectic Fillings

Wendl, Chris

doi:10.1007/978-3-319-91371-1_9

Cited by 1 publication

(3 citation statements)

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“…Therefore symplectic caps seemed to be on the boundary of symplectic rigidity and flexibility and it was unclear how restrictive caps are. Wendl [58] asked whether all contact structures admit symplectic caps. In the following result, we will show that this is indeed the case.…”

Section: Introductionmentioning

confidence: 99%

“…A weaker notion than Weinstein cobordism is that of a strong symplectic cobordism , which is exact near the boundary but perhaps not in the interior. Wendl [59] showed that symplectic caps are quite useful for constructing strong symplectic cobordisms between contact manifolds; for example, he showed that in dimension 3, there is a contact structure that is maximal for all contact manifolds with respect to strong cobordisms. Combining Wendl's argument with the existence of caps in Corollary 1.14, we can prove that there exists such a maximal contact structure in high‐dimension dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…Proof Following Wendl's argument [59], let

W^{2 n}

be a Liouville domain such that

\partial W

is disconnected with two components

false(Y_{1}, ξ_{1} false), false(Y_{2}, ξ_{2} false)

; such domains exist in all dimensions by [41]. There is a Weinstein cobordism

W_{1}

from

false(Y_{2}, ξ_{2} false) ∐ false(Y^{2 n - 1}, ξ false)

false(Y_{2}, ξ_{2} false) ♯ false(Y^{2 n - 1}, ξ false)

given by a Weinstein 1‐handle.…”

Section: Introductionmentioning

confidence: 99%

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Maximal contact and symplectic structures

Lazarev

2020

Journal of Topology

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We study the relationship on Weinstein domains given by Weinstein cobordism. Our main result is that any finite collection of high‐dimensional Weinstein domains with the same topology is Weinstein subdomains of a ‘maximal’ Weinstein domain also with the same topology. As applications, we construct many new exotic Weinstein structures, for example, exotic cotangent bundles containing many closed regular Lagrangians that are formally Lagrangian isotopic but not Hamiltonian isotopic and a new exotic Weinstein structure on Euclidean space. A novel feature of our construction of exotic structures is the use of results from symplectic flexibility. We also describe a similar construction in the contact setting which we use to produce ‘maximal’ contact structures and extend several existing results in low‐dimensional contact geometry to high dimensions. We prove that all contact manifolds have symplectic caps, introduce a general procedure for producing contact manifolds with many Weinstein fillings, and give a new proof of the existence of codimension two contact embeddings.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Proof Following Wendl's argument [59], let