Heegaard Floer homologies and contact structures

By Peter S. Ozsv'ath, Zolt'an Szab'o. 2002

93 citations recorded by the source corpus.

Given a contact structure on a closed, oriented three-manifold $Y$, we describe an invariant which takes values in the three-manifold's Floer homology $\HFa$. This invariant vanishes for overtwisted contact structures and is non-zero for Stein fillable ones. The construction uses of Giroux's interpretation of contact structures in terms of open book decompositions, and the knot Floer homologies introduced in math.GT/0209056.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0cb31f56d9c6db1ba39d41bd4b057c6e179e47b0

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In Computational Theory and Mathematics

1. A. Stipsicz
2. P. Ghiggini
3. Peter S. Ozsv'ath
4. Yi Ni
5. Zolt'an Szab'o
6. P. Lisca
7. M. Hedden
8. O. Plamenevskaya

In Discrete Mathematics and Combinatorics

1. A. Stipsicz
2. P. Ghiggini
3. P. Lisca
4. Peter S. Ozsv'ath
5. Zolt'an Szab'o
6. Yi Ni
7. O. Plamenevskaya
8. Ciprian Manolescu

In Geometry and Topology

1. A. Stipsicz
2. Peter S. Ozsv'ath
3. P. Lisca
4. Zolt'an Szab'o
5. M. Hedden
6. P. Ghiggini
7. Yi Ni
8. John A. Baldwin

Citations in the corpus

• On the existence of tight contact structures on Seifert fibered 3-manifolds (2007, background): “In [32] Ozsv´ath and Szab´o define an invariant, the contact Ozsv´ath–Szab´o invariant [22, 32].”
• Knot Floer homology detects fibred knots (2007, background): “When K is a fibred knot, it is shown that the topmost group of ̂ HFK(Y, K ) is a single Z [16].”
• Holomorphic disks and genus bounds (2003, methodology): “It is shown in [ 30 ] that this element is uniquely associated (up to sign) to the contact structure, i.e.”
• Knot Floer homology detects fibred knots (2006, background): “When K is a fibred knot, it is shown that the topmost group of ĤFK(Y,K) is a single Z [16].”
• Contact Structures with Distinct Heegaard Floer Invariants (2003, methodology): “Then FmixV,k (θ) = ±c(ξ).”
• Ozsvth-Szab invariants and tight contact 3-manifolds, III (2007, unknown): “A contact structure ξ on Y determines an element c(Y, ξ) ∈ ̂ HF(−Y, tξ) which has the following crucial properties [38]: • c(Y, ξ) is an isotopy invariant of the contact 3-manifold (Y, ξ); • c(Y, ξ) = 0 if the contact structure ξ is overtwisted; • c(Y, ξ) = 0 if (Y, ξ) is Stein fillable;”
• Holomorphic disks and knot invariants (2002, methodology): “e stated in Subsection 1.1. HOLOMORPHIC DISKS AND KNOT INVARIANTS 5 We will return to other applications of these constructions in future papers. For example, the constructions play a central role in [21], where they are used to define invariants of contact structures for three-manifolds. 1.3. Furthe”
• Bounds for the Thurston–Bennequin number from Floer homology (2003, methodology): “ants are the same as the Seiberg-Witten invariants of contact structures constructed in [KM]. The definition of c(ξ) uses an open book decomposition of the contact manifold; the reader is referred to [OS1] for the details. Acknowledgements I am grateful to Peter Kronheimer and Jake Rasmussen for ill”
• Planar open books and Floer homology (2005, unknown): “S G ] 20 A pr”
• Tight contact structures on some small Seifert fibered 3-manifolds (2005, background): “In a remarkable series of papers [24, 25, 26, 30] Ozsváth and Szabó defined new invariants of many low–dimensional objects — including contact structures on closed 3–manifolds.”
• Heegaard Floer homology and alternating knots. (2002, methodology): “Thus, according to Theorem 1.4 of [ 26 ], the induced contact structure is tight.”