A cylindrical reformulation of Heegaard Floer homology

By Robert Lipshitz. 2005

212 citations recorded by the source corpus.

We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold U a0;1c R, where U is the Heegaard surface, instead of Sym g .U/. We then show that the entire invariance proof can be carried out in our setting. In the process, we derive a new formula for the index of the @‐operator in Heegaard Floer homology, and shorten several proofs. After proving invariance, we show that our construction is equivalent to the original construction of Ozsvath‐Szabo. We conclude with a discussion of elaborations of Heegaard Floer homology suggested by our construction, as well as a brief discussion of the relation with a program of C Taubes.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8bb12735f99e6e2b5f651ea9ea76d417ec9ea514

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Grid Homology for Knots and Links (2015, methodology): “This construction was proposed and studied by Lipshitz [121]; see also [120] for Lipshitz’s corresponding holomorphic construction.”
A combinatorial description of knot Floer homology (2006, methodology): “Given φ ∈ π2(x,y), Lipshitz gives a formula for its Maslov index μ(φ) in terms of combinatorial data on the Heegaard surface [12].”
A connected sum formula for involutive Heegaard Floer homology (2016, methodology): “The following lemma is phrased in terms of the “cylindrical reformulation” of Heegaard Floer homology developed by Lipshitz in [13].”
Link cobordisms and functoriality in link Floer homology (2016, background): “lity, which is described in [OS04a, Def. 4.10], and ensures necessary niteness results on counts of owlines. Given an almost complex structure J s on [0;1] R which is suitably generic in the sense of [Lip06], we dene an endomorphism @: CFL(H;s) !CFL(H;s) by the formula @(x) = X y2T 2\T X ˚2ˇ(x;y) (˚”
Quasi-stabilization and basepoint moving maps in link Floer homology (2016, background): “es transversality for Maslov index 2 holomorphic -degenerate disks ([OS04b, Prop. 3.14]). In the cylindrical setup, if a sequence of holomorphic strips for the almost complex structures considered in [Lip06] has a cylindrical boundary degeneration in its limit, the boundary degeneration will be j j ”
Holomorphic disks, link invariants and the multi-variable Alexander polynomial (2008, methodology): “of this moduli space by the action of R. Also, µ(φ) denotes the expected dimension of M(φ) (i.e. the Maslov index of φ). (Indeed, we will ﬁnd it useful later to use Lipshitz’s cylindrical formulation [14] instead; we return to this point in Section 5.) We will establish the following in Subsection 5”
Connected sums and involutive knot Floer homology (2017, methodology): “Equivalently, one can count holomorphic curves inside of Σ × [0, 1] × R using Lipshitz’s cylindrical reformulation [21].”
HF=HM, II : Reeb orbits and holomorphic curves for the ech/Heegaard Floer correspondence (2010, methodology): “ × Σ writes this symplectic form as ds ∧ dt + w Σ . Part 2: The identification M (1,2) = (1, 2) × Σ identifies the geometry just defined with the geometry that Lipshitz introduces in the Section 1 of [L]. Lipshitz introduces in this section of his paper certain sorts of almost complex structure on R”
An algorithm for computing some Heegaard Floer homologies (2006, unknown): “See [14], [13], [16], [3] for details.”
Bordered Heegaard Floer homology (2008, methodology): “23 is a simple adaptation of the techniques for closed 3-manifolds [33] (as modi ed for the cylindrical case in [20], say).”
Heegaard Floer homology and integer surgeries on links (2010, unknown): “Here, np(D) denotes the average multiplicity of D in the four quadrants around a point p, while e(D) is the Euler measure of the domain, as defined in [8].”
Link Floer homology detects the Thurston norm (2006, background): “(This correspondence can be seen through the cylindrical reformulation of Heegaard Floer homology due to Lipshitz [7].”
Moving basepoints and the induced automorphisms of link Floer homology (2011, methodology): “'/j D 1, following standard practice, we will use Lipshitz’s cylindrical reformulation [4]; in the cylindrical version, we will count the number of holomorphic embeddings of surfaces F ,!T fz 2C j jzj 1; z(3)¤ 1g satisfying certain conditions, such as that the image of the projection onto the first ”
On the equivalence of Legendrian and transverse invariants in knot Floer homology (2011, background): “This disk has Maslov index one if no other components of y are contained in its domain, and greater than one otherwise (see Lipshitz’s formula for the Maslov index in [20]).”
The contact invariant in sutured Floer homology (2007, methodology): “corners. (All other domains nontrivially THE CONTACT INVARIANT IN SUTURED FLOER HOMOLOGY 19 a h(a) + − x x y h(b) Figure 13. intersect the suture Γ.) We now apply Lipshitz’ formula (Corollary 4.10 of [Li1]) µ(Di) = n x(Di)+n y(Di)+e(Di), for computing the index of Di. Here e(Di) is the Euler measure”