Few remarks about symplectic filling
By Y. Eliashberg. 2003
154 citations recorded by the source corpus.
We show that any compact symplectic manifold $W,\omega$ with boundary embeds as a domain into a closed symplectic manifold, provided that there exists a contact plane $\xi$ on $\partial W$ which is weakly compatible with $\Omega$, i.e. the restriction $\omega|_\xi$ does not vanish and the contact orientation of $\partial W$ and its orientation as the boundary of the symplectic manifold $W$ coincide. This result provides a useful tool for new applications of Ozsv\'ath-Szab\'o and Seiberg-Witten Floer homology theories in 3-dimensional topology.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e0276c4dd9a9e38bc2f57490ced9ca3b5e9ccca3
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Citations in the corpus
- 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.”
- 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].”
- 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].”
- Planar open books and Floer homology (2005, result): “More visibly, it is the inspiration for the recently proved embedding result of Eliashberg [3] and Etnyre [6] leading — among other results — to the proof that nontrivial knots have Property P [13].”
- Strongly fillable contact 3-manifolds without Stein fillings (2005, background): “Remark In the literature there are several different equivalent definitions of Stein manifold: see for example [ 4, Section 4 ].”
- 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 + > 0. Then, according to [8, 30] ξ is not planar.”
- 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, background): “By contrast, most other constructions of symplectic cobordisms from contact 3–manifolds to contact 3–manifolds (as in Etnyre and Honda [11]) are built out of elementary 0–, 1– and 2–handle cobordisms as in Eliashberg [7] or Weinstein [25].”
- Foliations and the Geometry of 3-Manifolds (2007, background): “Remark Recently, Eliashberg [66] has shown by an explicit construction that every taut foliation F except for the product foliation on S2× S1 is symplectically fillable.”
- Dehn surgery, the fundamental group and SU(2) (2003, methodology): “The following proposition is proved in [13] using the results of [7] and [6]:”
- Holomorphic disks and genus bounds (2003, background): “[ 3 ], restated in Theorem 4.1 below) any weakly semi-llable contact structure is weakly llable, as well.”
- 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).”
- Ozsváth-Szabó invariants and fillability of contact structures (2004, methodology): “Using methods from [7] one can prove that −Σ(2, 3, 17) admits at most three tight contact structures up to isotopy, therefore Proposition 3.”
- Lectures on open book decompositions and contact structures (2004, background): “Theorem 5.1 (Eliashberg 2004 [6]; Etnyre 2004 [14]).”