On symplectic fillings
By John B. Etnyre. 2003
89 citations recorded by the source corpus.
In this note we make several observations concerning symplectic fillings. In particular we show that a (strongly or weakly) semi-fillable contact structure is fillable and any filling embeds as a symplectic domain in a closed symplectic manifold. We also relate properties of the open book decomposition of a contact manifold to its possible fillings. These results are also useful in showing the contact Heegaard Floer invariant of a fillable contact structure does not vanish [28] and property P for knots [18].
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/aab8d89d5e1bdb7125ba53de047f5f7b356b7a37
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Citations in the corpus
- Planar open books and Floer homology (2005, result): “Theorem 1.1 (Etnyre, [7]).”
- Ozsváth-Szabó invariants and tight contact three-manifolds, II (2004, methodology): “By [5, 9 ] one can use a symplectic cap to construct a symplectic filling of (M, ξ) with b+2 > 0. Then, according to [8, 30] ξ is not planar.”
- Khovanov module and the detection of unlinks (2012, background): “Eliashberg [5] shows that the weak semi-filling Y × [−1, 1] can be embedded in a closed symplectic 4–manifold X (see also Etnyre [7], for an alternative construction).”
- Tight contact structures on some small Seifert fibered 3-manifolds (2005, background): “By [4, 5] we can cap off n of the boundary components of X̃ with symplectic caps having b2 > 0, obtaining a symplectic filling of (Y, ξ) with b2 > 0, which is against our assumptions.”
- Four-dimensional symplectic cobordisms containing three-handles (2006, methodology): “M 0; 0/, we produce a contact structure which is isotopic to 0 , using the fact (see Geiges [13]) that full Lutz twists do not change homotopy classes of contact structures and the fact (Eliashberg [5]) that homotopic, overtwisted contact structures are isotopic.”
- Knot Floer homology and rational surgeries (2005, background): “[4] and [5]), it is now possible to prove results of this type purely in the context of Heegaard Floer homology, see also [21].”
- Ozsvth-Szab invariants and tight contact 3-manifolds, III (2007, methodology): “By [5, 9] one can use a symplectic cap to construct a symplectic filling of (M, ξ) with b2 > 0.”
- Reeb vector fields and open book decompositions (2008, background): “It was also ope n book decompositions that enabled the construction of concave symplectic fillings for any contact 3-manifold (due to Eliashberg [El] and Etnyre [ Et1 ]); this in turn was the missing ingredient in Kronheimer-Mrowka’s proof of Property P for knots [KM].”
- Ozsváth-Szabó invariants and fillability of contact structures (2004, unknown): “By [4], Lemma 3.”
- 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].”
- Lectures on open book decompositions and contact structures (2004, background): “1 (Eliashberg 2004 [6]; Etnyre 2004 [14]).”
- Holomorphic disks and genus bounds (2003, background): “The nal breakthrough which makes this paper possible is an embedding theorem of Eliashberg [3], see also [ 6 ] and [25], which shows that a symplectic semi-lling of a three-manifold can be embedded in a closed, symplectic four-manifold.”
- Link Floer homology detects the Thurston norm (2006, background): “Let f : Y → [0, 3 ] be a Morse function corresponding to the Heegaard diagram, ∇f is the gradient vector field associated to f. Let w be the union of the flowlines of ∇f, such that each of these flowlines passes through a point in w, and connects an index 0 critical point to an index 3 critical poin”