Rounding corners of polygons and the embedded contact homology of T 3

By Michael Hutchings, Michael C. Sullivan. 2004

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The embedded contact homology (ECH) of a 3‐manifold with a contact form is a variant of Eliashberg‐Givental‐Hofer’s symplectic field theory, which counts certain embedded J ‐holomorphic curves in the symplectization. We show that the ECH of T 3 is computed by a combinatorial chain complex which is generated by labeled convex polygons in the plane with vertices at lattice points, and whose differential involves “rounding corners”. We compute the homology of this combinatorial chain complex. The answer agrees with the Ozsvath‐Szabo Floer homology HF C .T 3 /.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e3812e90c47f3ab86abef1726e06a01ca146f701

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Citations in the corpus

Heegaard Floer homology and genus one, one‐boundary component open books (2008, background): “Embedded contact homology associates a graded Abelian group ECH∗(Y, λ; Γ) to a closed oriented 3-manifold Y , a contact form λ, and a homology class Γ ∈ H1(Y ) [14].”
Beyond ECH capacities (2014, result): “One can then “round corners” as in [12, Section 1.3.3] to reduce the number of lattice points to what it was before; this decreases A (cid:127) similarly to [12, Lemma 2.14].”
Sutures and contact homology I (2010, background): “In contrast to Corollary 1.12, the Legendrian contact homology — due to Chekanov [Ch] and Eliashberg in dimension three, and Ekholm-Etnyre-Sullivan [EES] in higher dimensions — of a stabilized Legendrian submanifold always vanishes.”
The periodic Floer homology of a Dehn twist. (2004, unknown): “Suppose that our symplectomorphism φ has an invariant curve ξ ⊂ Σ, on which φ is an irrational rotation, and which divides Σ into pieces Σ1 and Σ2.”
Symplectic embeddings into four‐dimensional concave toric domains (2013, background): “5) One can also define a combinatorial differential on the chain complex ECC ∗ (Ω) similarly to [8], which agrees with the ECH differential for appropriate perturbations of the contact form and almost complex structures, but we do not need this here.”
Quantitative embedded contact homology (2010, methodology): “One can calculate the ECH capacities of a polydisk by understanding the ECH chain complex of an appropriately smoothed polydisk, similarly to the calculations in [8] for T 3 as outlined in §6.”
The Seiberg–Witten equations and the Weinstein conjecture (2006, unknown): “This planned sequel will identify a version of the Seiberg–Witten Floer homology for a given compact, oriented 3–manifold with a variant of the Eliashberg–Givental–Hofer contact homology [7], a variant along the lines of Hutchings’ embedded contact homology (see Hutchings and Sullivan [14]).”
The embedded contact homology index revisited (2008, methodology): “See [13] for more about ECH, and [14, §7] for a proof that ∂2 = 0.”
HF=HM, I : Heegaard Floer homology and Seiberg–Witten Floer homology (2010, background): “The definition of twisted embedded contact homology in [4] requires the choice of a subgroup of H2(Y;Z).”
Embedded contact homology and Seiberg–Witten Floer cohomology II (2008, background): “The first four papers [10; 11; 12; 13] in this series construct an isomorphism between Hutchings’ embedded contact homology [3; 4; 5] and the Seiberg–Witten Floer cohomology as defined by Kronheimer and Mrowka in [7].”
Lecture Notes on Embedded Contact Homology (2013, background): “The homology of the combinatorial differential δ is computed in [28] (there with Z coefficients), and the conclusion (with Z/2 coefficients) is that”
Taubes’s proof of the Weinstein conjecture in dimension three (2009, background): “For the full definition of this theory, see [33] for an overview and [32, 35] for more details.”
Relative asymptotic behavior of pseudoholomorphic half‐cylinders (2007, background): “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”
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].”