Holomorphic triangles and invariants for smooth four-manifolds

By Peter S. Ozsv'ath, Zolt'an Szab'o. 2001

378 citations recorded by the source corpus.

Abstract In earlier articles, the authors introduced invariants for closed, oriented three-manifolds, defined using a variant of Lagrangian Floer homology in the symmetric products of Riemann surfaces. The aim of this article is to introduce invariants of oriented, smooth four-manifolds, built using these Floer homology groups. This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology groups.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dbdc298c0a666d93435059096cde1a2c19fbef54

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In Applied Mathematics

In Geometry and Topology

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Citations in the corpus

Contact structures, sutured Floer homology and TQFT (2008, positive): “tible partial open book decomposition (S,R+(Γ),h). 3. EXPLANATION OF THEOREM 1.1 3.1. Naturality. We now explain what we mean by a “natural map” Φ ξ. Recall the following theorem from Ozsva´th-Szabo´ [OS7]: Theorem 3.1 (Ozsva´th-Szabo´). Given two Heegaard decompositions (Σ,α,β), (Σ,α,β) of a closed”
Connected sums and involutive knot Floer homology (2017, methodology): “If s is torsion, then an absolute lift of grw can be specified using the absolute grading on CF∞(Y,w, s) from [29].”
On the spectral sequence from Khovanov homology to Heegaard Floer homology (2008, neutral): “Similarly, the double cover of S 3 × [0, 1] branched along Z is a 4-dimensional cobordism from Σ(L1) to Σ(L2), and, therefore, induces a map from ĤF (−Σ(L1)) to ĤF (−Σ(L2)) [19].”
Khovanov module and the detection of unlinks (2012, neutral): “The map on Floer homology induced by a cobordism is defined by associating chain maps to handle attachments in a handle decomposition [29].”
Floer homology and knot complements (2003, methodology): “in [12]. They possess many of the known or expected properties of the Seiberg-Witten Floer homologies, and have been used to deﬁne four-manifold invariants analogous to the Seiberg-Witten invariants ([21], [20]). It would be a mistake, however, to think of the Ozsv´ath-Szab´o theory as being nothing”
Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture (2013, background): “Both monopole Floer homology, as constructed by Kronheimer and Mrowka in their book [32], and the Heegaard Floer homology of Ozsváth and Szabó [46–48] are defined for arbitrary 3-manifolds.”
Strongly fillable contact 3-manifolds without Stein fillings (2005, neutral): “We will not treat the subject in its most general form, but only in the form it will be needed in the proof of Theorem 2.4.”
Knot Floer homology detects genus-one fibred knots (2006, background): “ection we will give a brief overview of the results in Heegaard Floer theory we will need in the following, with no pretension of completeness. The details can be found in Ozsvath and Szabo´’s papers [16, 15, 20, 14, 19, 13]. 2.1 Heegaard Floer homology Let Y be a closed, connected, oriented 3–manif”
Planar open books and Floer homology (2005, background): “/destabilizations. 2.2. Heegaard Floer homologies. Let W be a connected, oriented four-manifold with two boundary components, ∂W = −Y 1 ∪ Y 2. Then, we denote this cobordism by W : Y 1 −→ Y 2. Recall [18] that there is an induced Z[U]–equivariant map F+ W : HF +(Y 1) −→ HF+(Y 2) on Heegaard Floer ho”
Tight contact structures on some small Seifert fibered 3-manifolds (2005, background): “Since F+ −X,s is a homomorphism of Z[U]– modules [ 26 ], this immediately implies that F+ −X,s restricted to HF”