Planar open book decompositions and contact structures

By John B. Etnyre. 2004

147 citations recorded by the source corpus.

We observe that while all overtwisted contact structures on compact 3-manifolds are supported by planar open book decompositions, not all contact structures are. This has relevance to the Weinstein conjecture and invariants of contact structures.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2f3fe986fb3ac34ac52f378e66d5f4a1e35289a9

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Lectures on open book decompositions and contact structures (2004, background): “Then there is a one to one correspondence between {oriented contact structures on M up to isotopy} and {open book decompositions of M up to positive stabilization}.”
Ozsváth-Szabó invariants and tight contact three-manifolds, II (2004, methodology): “0;r1,r2,r3), e0 ≥ −1 and M is an L–space, then each contact structure on M is planar and therefore satisﬁes Weinstein’s conjecture. 25 Proof It is known that overtwisted contact structures are planar [8], hence we can focus on tight structures only. For small Seifert ﬁbered rational homology 3–spher”
Planar open books and Floer homology (2005, background): “In [7], Etnyre gives the following constraints on contact structures compatible with planar open book decompositions.”
Planar open books, monodromy factorizations and symplectic fillings (2009, unknown): “Acknowledgements We are grateful to John Etnyre for some helpful conversations, to Dan Margalit and Chris Wendl for helpful correspondence, and to Paolo Lisca and András Stipsicz for their comments on a preliminary version of this paper.”
Cabling, contact structures and mapping class monoids (2010, background): “There are many techniques for putting Legendrian knots on pages of open book decompositions [16, 22] (though this is still more of an art than a science).”
Weak and strong fillability of higher dimensional contact manifolds (2011, background): “This is a serious constraint, as there are many smooth manifolds that are known to admit contact structures but not Liouville pairs: for instance, by [19] and [1], this is the case for any 3-manifold whose contact structures are all known to be planar (e.”
Reeb vector fields and open book decompositions (2008, methodology): “Prior to the work of Taubes, the Weinstein conjecture in dimension three was verified for contact structures which admit planar open book decompositions [ACH] (also see related work of Etnyre [ Et2 ]), for certain Stein fillable contact structure s [Ch, Ze], and for certain universally tight contact”
A hierarchy of local symplectic filling obstructions for contact $3$-manifolds (2010, result): “to realize (M,ξ) as a non-separating contact type hypersurface. Corollary 4 also generalizes similar results proved by McDuﬀ for the tight 3-sphere [McD91] and Etnyre for all planar contact manifolds [Etn04]. The algebraic counterpart to Corollary 4 involves the so-called U-map in Embedded Contact H”
THE WEINSTEIN CONJECTURE FOR PLANAR CONTACT STRUCTURES IN DIMENSION THREE (2004, unknown): “Let ˜ J be a compatible almost complex structure on R × M .”
Ozsvth-Szab invariants and tight contact 3-manifolds, III (2007, background): “Then, according to [8, 31] ξ is not planar.”