On symplectic cobordisms

By John B. Etnyre, K. Honda. 2001

117 citations recorded by the source corpus.

Abstract. In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are “symplectic cobordism equivalent”.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/52127a79f3d11d7d722f52db600dace06ec51f8c

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Citations in the corpus

Making cobordisms symplectic (2015, background): “A similar result for overtwisted contact 3-manifolds is proven in [9], and for general contact 3-manifolds in [6] and [8].”
A hierarchy of local symplectic filling obstructions for contact $3$-manifolds (2010, result): “not strongly ﬁllable. Though our proof of non-ﬁllability will not depend on it, the implication that (M,ξ) is not strongly ﬁllable follows from the above statement due to a result of Etnyre and Honda [EH02], that every contact 3-manifold is concave ﬁllable: this means that strong ﬁllings can always ”
Ozsváth-Szabó invariants and fillability of contact structures (2004, unknown): “be a contact manifold, then there exists a concave symplectic ﬁlling (W′,ω W′) of (Y,ξ) with canonical Spinc-structure k W′ such that b+ 2 (W ′) > 1 and c+(ξ) = Fmix W′,k W′ (Θ−). Proof. Combining [6], Theorem 1.1 and [5] Lemma 3.1 there is a Stein ﬁllable contact manifold (Y′,ξ′) and a symplecti”
Algebraic Torsion in Contact Manifolds (2010, background): “ 3-manifolds with planar torsion. Remark 1.6. Sometimes exact cobordisms are known to exist when the negative end has a smaller order of algebraic torsion than the positive end, e.g. Etnyre and Honda [EH02] have shown that any positive end is allowed if the negative end is overtwisted (meaning 0-tor”
Four-dimensional symplectic cobordisms containing three-handles (2006, background): “John Etnyre pointed out to the author that the same argument together with compactness results for pseudo-holomorphic curves implies that any contact three-manifold of non-zero torsion satisfies the Weinstein conjecture.”
Holomorphic disks and genus bounds (2003, background): “Although Eliashberg’s is the construction we need, concave fillings have been constructed previously in a number of different contexts, see for example [22], [1], [7], [10], [25].”
Contact 3-manifolds with infinitely many Stein fillings (2003, result): “We only sketch the proof of the above theorem, since it was already proved in [14], [3] (with the extra assumption of having only homologically essential vanishing cycles in the fibration), and also appeared in [7].”
A Legendrian surgery presentation of contact 3-manifolds (2001, unknown): “5], cf. [15] and Section 3 below. In particular, if (M′,ξ′) is obtained from (M,ξ) by contact (−1)–surgery along a Legendrian link, then there is a socalled Stein cobordism from (M,ξ) to (M′,ξ′), see [7] for details, and [6] for background information. This is denoted by (M,ξ) ≺ (M′,ξ′). If (S3,ξ0) ”
Few remarks about symplectic filling (2003, methodology): “His idea is that by performing a sequence of Legendrian contact surgeries it is possible to transform a contact manifold into a homology sphere and thus, taking into account an argument from Proposition 4.1, to reduce the problem to the case considered in their paper [ 12 ] with Ko Honda.”
Surgery diagrams for contact 3-manifolds (2003, background): “We leave it as an exercise to the reader to construct such K1 (see the examples in [12] and [9]); that even integers are excluded follows from [6, Prop.”
Birational cobordism invariance of uniruled symplectic manifolds (2006, background): “t. There is a related notion in dimension 4 in [OO]. However we remark that the cobordism relation studied in this paper is quite diﬀerent from some other notions of symplectic cobordisms, see [EGH], [EH], [Gi], [GGK]. According to [GS], we have the following basic factorization result. Theorem 2.2.”
Lectures on open book decompositions and contact structures (2004, background): “14 (Etnyre and Honda 2002, [16]; Gay 2002, [19]).”
Cabling, contact structures and mapping class monoids (2010, background): “Using results from [10, 19, 47] we have the following corollary.”