An index inequality for embedded pseudoholomorphic curves in symplectizations

By Michael Hutchings. 2001

160 citations recorded by the source corpus.

Abstract.Let Σ be a surface with a symplectic form, let φ be a symplectomorphism of Σ, and let Y be the mapping torus of φ. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in ℝ×Y, with cylindrical ends asymptotic to periodic orbits of φ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces.¶This paper establishes some of the foundations for a program with Michael Thaddeus, to understand the Seiberg-Witten Floer homology of Y in terms of such pseudoholomorphic curves. Analogues of our results should also hold in three dimensional contact topology.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4008334b7b52ebd0a470da4ffe76c32a20d6ddc0

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Citations in the corpus

Lecture Notes on Embedded Contact Homology (2013, methodology): “The definition of the ECH index is the key nontrivial part of the definition of ECH; the original references are [21, 22], and we will spend considerable time explaining this in §3.”
The periodic Floer homology of a Dehn twist. (2004, methodology): “Under “monotonicity” and “d-regularity” assumptions on φ, we can define the periodic Floer homology (PFH), denoted by HP∗(φ, h); see [12], the background in [9], and the review in Section 2.”
Proof of the Arnold chord conjecture in three dimensions, II (2010, unknown): “Θ′ with [Θ] = [Θ′] = Γ, the diﬀerential coeﬃcient h∂Θ,Θ′i ∈ Z/2 is deﬁned to be the mod 2 count of J-holomorphic curves in MJ(Θ,Θ′) with “ECH index” equal to 1. For the deﬁnition of the ECH index see [4, 5]. If Jis generic, then ∂is well-deﬁned and ∂2 = 0, as 3 shown in [10, §7]. A symplectization-a”
Gluing pseudoholomorphic curves along branched covered cylinders II (2007, result): “h intersection point counts positively. Proof. Part (a) is proved in [10, Eq. (18) and Prop. 6.1], and part (b) is proved in [10, Prop. 7.1], except for two issues. First, these results are proved in [10] in a slightly diﬀerent setting where Y is a mapping torus and an analytical simplifying assumpt”
A survey of Heegaard Floer homology (2013, background): “Surprisingly, the equivalence between HM and HF passes through embedded contact homology (ECH), defined by Hutchings [31] and Hutchings-Taubes [32, 33].”
Embedded contact homology and Seiberg–Witten Floer cohomology II (2008, background): “As explained to the author by Mike Hutchings, the following is a consequence of the index inequalities in [1]:”
HF=HM, II : Reeb orbits and holomorphic curves for the ech/Heegaard Floer correspondence (2010, methodology): “mber of segments that comprise the M δ part of the union of the integral curves of v from any given Θ ∈ Zˆ ech,M . The versions here of d and genus Σ are equal, but Assertion (a)(i) of Theorem 1.8 of [Hu3] as stated requires d > genus(Σ). Even so, the I ech = 1 and d = genus(Σ) version of Asserti”
Sutures and contact homology I (2010, methodology): “[Hu1] M. Hutchings,An index inequality for embedded pseudoholomorphic curvesin symplectizations, J. Eur.”
The embedded contact homology index revisited (2008, background): “17 in [11] assumed that neither of the positive ends of C corresponding to ζi or ζj is “trivial”, i.”
Intersection theory of punctured pseudoholomorphic curves (2009, methodology): “We also prove an adjunction formula which gives a topological condition that will guarantee a curve in a given homotopy class is embedded, extending previous work of Hutchings from [14].”
A hierarchy of local symplectic filling obstructions for contact $3$-manifolds (2010, background): “,λ,J;h) := ECH]∗(M,λ,J;h,H2(M)). At the other end of the spectrum, taking Gto be the trivial subgroup leads to the fully twisted variant of ECH, ECH] ∗(M,λ,J;h) := ECH]∗(M,λ,J;h,{0}). 8The results in [Hut02] are stated only for a very special class of stable Hamiltonian structures arising from mappi”
Symplectic Floer homology of area-preserving surface diffeomorphisms (2008, methodology): “Hutchings and Sullivan have an argument involving intersection positivity with the J-holomorphic foliation coming from R×T for T a torus in Mφ which is invariant under the mapping torus flow [HS1, Lemma 3.11].”
Cylindrical contact homology for dynamically convex contact forms in three dimensions (2014, unknown): “This is proved similarly to the relative adjunction formula in [16, Rmk. 3.2].”
Rounding corners of polygons and the embedded contact homology of T 3 (2004, methodology): “) The main result of [7] shows that if Memb(α, β, Z)C denotes the component of Memb(α, β, Z) containing the J-holomorphic curve C, then dim ( Memb(α, β, Z)C ) ≤ I(α, β, Z), (102)”
Relative asymptotic behavior of pseudoholomorphic half‐cylinders (2007, methodology): “This intersection theory in turn is important for the foundations of a 4-dimensional variant of symplectic field theory due to Hutchings known as embedded contact homology [9, 10, 11] - a theory which exploits the fact that the embeddedness of punctured curves in dimension 4 can be controlled in ter”