Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary

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

571 citations recorded by the source corpus.

Abstract In Ozsvath and Szabo (Holomorphic triangles and invariants for smooth four-manifolds, math. SG/0110169, 2001), we introduced absolute gradings on the three-manifold invariants developed in Ozsvath and Szabo (Holomorphic disks and topological invariants for closed three-manifolds, math.SG/0101206, Ann. of Math. (2001), to appear). Coupled with the surgery long exact sequences, we obtain a number of three- and four-dimensional applications of this absolute grading including strengthenings of the “complexity bounds” derived in Ozsvath and Szabo (Holomorphic disks and three-manifold invariants: properties and applications, math.SG/0105202, Ann. of Math. (2001), to appear), restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF + for a variety of three-manifolds. Moreover, we show how the structure of HF + constrains the exoticness of definite intersection forms for smooth four-manifolds which bound a given three-manifold. In addition to these new applications, the techniques also provide alternate proofs of Donaldson's diagonalizability theorem and the Thom conjecture for CP 2 .

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

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Citations in the corpus

On the existence of tight contact structures on Seifert fibered 3-manifolds (2007, background): “…group HF ( Y, t ) comes with a natu-ral relative Z –grading. d Thus, when c 1 ( t ) is torsion the Ozsv´ath–Szab´o homology group HF ( Y, t ) splits as Moreover, this relative Z –grading admits a natural lift to an absolute Q –grading [30]. where the degree d 0 ∈ Q is determined mod 1 by t .”
An infinite-rank summand of topologically slice knots (2013, methodology): “Building upon the techniques of Ozsváth and Szabó [29], Khovanov [22] and Rasmussen [34], Livingston’s Z–summand result can be improved to a Z –summand (see Livingston [27] and Manolescu and Owens [28]), using the smooth concordance homomorphisms given by the Rasmussen s invariant, and the Manolescu”
Lens space surgeries and L-space homology spheres (2007, background): “Although ĤF (K1) ∼= ĤF (K ′ 1), the two spaces can be distinguished by their d-invariants [25].”
On Knot Floer Homology and Cabling: 2 (2008, result): “To this end, our results provide the following obstructions: Corollary 1.6.”
The knot Floer complex and the smooth concordance group (2011, background): “quot;(K0). For completeness, we sketch the proof below. Sketch of proof. To prove (1), we consider the d-invariants of large surgery along K. If Kis slice, then the surgery correction terms dened in [OS03a] vanish, i.e., agree with the surgery correction terms of the unknot, and the maps HFd(S3 N(K)”
Tight contact structures on some small Seifert fibered 3-manifolds (2005, background): “The Ozsváth–Szabó homology groups in this triangle are well–known (see [27]):”
Connected sums and involutive knot Floer homology (2017, methodology): “They define two correction terms, d and d, analogous to the Ozsváth-Szabó d invariant from Heegaard Floer homology [25].”
Strongly fillable contact 3-manifolds without Stein fillings (2005, methodology): “The group HF +(−�(2,3,6n + 5)) is computed in [ 17, Section 8 ]. From this it is easy to prove that d HF (+1)(�(2,3,6n + 5)) is isomorphic to (Z/2Z)n by applying the exact triangle (1) and the isomorphism d HF (d)(Y ) ∼= d”
Concordance homomorphisms from knot Floer homology (2014, background): “This follows quickly from the behaviour of the maps associated to negative definite cobordisms, see [23].”
Contact surgeries and the transverse invariant in knot Floer homology (2008, background): “For more on the absolute (rational) Maslov grading on ĈF(Y, t) where c1(t) is a torsion class, see [18].”
Grid Homology for Knots and Links (2015, methodology): “After this breakthrough, a number of other proofs have emerged, using Seiberg-Witten theory [107, 150] and Heegaard Floer homology [168].”
L-space surgeries, genus bounds, and the cabling conjecture (2010, methodology): “This fact holds in the setting of Heegaard Floer homology as well [25], the framework in place for the remainder of this paper.”
Topologically slice knots of smooth concordance order two (2012, methodology): “The following theorem from [30] provides the obstruction we will use to show that knots are not smoothly slice.”