Symplectic fillability of tight contact structures on torus bundles

By Fan Ding, Hansjörg Geiges. 2001

86 citations recorded by the source corpus.

We study weak versus strong symplectic llability of some tight contact structures on torus bundles over the circle. In particular, we prove that almost all of these tight contact structures are weakly, but not strongly symplectically llable. For the 3{torus this theorem was established by Eliashberg. AMS Classication 53D35; 57M50, 57R65

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/51f4cc1d0e763525b85e05f87d2c6c779b54254e

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In Geometry and Topology

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Citations in the corpus

The vanishing of the contact invariant in the presence of torsion (2007, background): “Proof. If (N ′, ξ′) is obtained from (N, ξ) by contact (+1)-surgery, then (N, ξ) is obtained from (N ′, ξ′) by contact (−1)-surgery (i.e., Legendrian surgery); see [ DG1 , Proposition 8].”
Tight contact structures on some small Seifert fibered 3-manifolds (2005, background): “(For more about contact surgery see [1, 2, 3].”
On the classification of tight contact structures (2002, background): “More generally one can define Legendrian (r) surgery, for r ∈ Q, as described in [3].”
Knot Floer homology detects genus-one fibred knots (2006, background): “Given a contact manifold (Y , ξ) and a Legendrian knot K ⊂ Y there is an operation called contact ( + 1)-surgery which produces a new contact manifold (Y ′, ξ′); see [2] and [3].”
Four-dimensional symplectic cobordisms containing three-handles (2006, unknown): “M; / contains .”
Ozsváth-Szabó invariants and tight contact three-manifolds, II (2004, methodology): “The purpose of the present paper is to show that contact Ozsváth–Szabó invariants [28] can be effectively combined with contact surgery [4, 5] to tackle problem (P).”
Tight contact structures with no symplectic fillings (2000, background): “This result was recently extended by Ding-Geiges [ 6 ] to T 2 -bundles over S 1 . It is not known whether the other inclusions in the diagram are strict.”
A Legendrian surgery presentation of contact 3-manifolds (2001, background): “If (S3, ξ0) ≺ (M ′, ξ′), then (M ′, ξ′) is Stein fillable; the concave end (S3, ξ0) of the Stein cobordism can be filled with the standard symplectic 4–ball.”
On symplectic fillings (2003, background): “The properness of the inclusion of the set of strongly fillable structures into the set of weakly fillable structures is due to Eliashberg [4] (for further examples see [2]).”
On the existence of tight contact structures on Seifert fibered 3-manifolds (2007, methodology): “By [3, 4], this diagram is equivalent to a contact surgery diagram as in Figure 7.”
Ozsváth-Szabó invariants and fillability of contact structures (2004, background): “The first example of a weakly but not strongly fillable contact manifold was discovered on T 3 by Eliashberg [2], and more examples were constructed by Ding and Geiges [1] on torus bundles over S(1) building on Eliashberg’s.”
Handle moves in contact surgery diagrams (2008, background): “Introduction In [3] and [4] we introduced the notion of contact Dehn surgery, especially contact (±1)surgery; see also [16, Section 6.”
Right-veering diffeomorphisms of compact surfaces with boundary (2005, background): “are not strongly fillable were first exhibited by Eliashberg [El2]; further examples were given by Ding–Geiges [ DG ].”
Strongly fillable contact 3-manifolds without Stein fillings (2005, background): “A weakly fillable but non Strongly fillable contact structure has been found first by Eliashberg [3] and later more have been found by Ding and Geiges [1].”