# Embedded contact homology and applications

Here is a talk from a few years ago which gives an overview of ECH and some of its older applications:
• Applications of ECH, Freedman birthday conference, June 8, 2011. slides video.
To get started reading about ECH, see the lecture notes [15] below.
• [1] An index inequality for embedded pseudoholomorphic curves in symplectizations
Journal of the European Math. Soc. 4 (2002), 313-361. PDF PS
This paper defines the index, or grading, in PFH. When Y is the mapping torus of a surface symplectomorphism, this index associates to each pseudoholomorphic curve C in R x Y an integer I(C), which depends only on the relative homology class of C. One can think of I(C) as a kind of measure of the complexity of C, which is quite different from usual measures of complexity such as genus or number of ends. In particular, we show that for a generic almost complex structure, there are no curves in R x Y with I<0; a curve has I=0 if and only if it is a cover of a union of R-invariant cylinders; I=1 curves are (mostly) embedded, have Fredholm index 1, and satisfy additional topological constraints; and there are good compactness results for moduli spaces of I=1 and I=2 curves. An analogous story holds for ECH; see paper [3] below. The differential in PFH or ECH counts curves with I=1. The index I is the key, nontrivial part of the definition of PFH and ECH.

• [2] The periodic Floer homology of a Dehn twist (with M. Sullivan),
This paper reviews the definition of PFH and computes the PFH of some Dehn twists. Some of the geometric techniques introduced here play an important role in paper [3] below.

• [3] Rounding corners of polygons and the embedded contact homology of T^3 (with M. Sullivan),
This paper gives an introduction to ECH, and computes the ECH of T^3. The latter is described 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". Eli Lebow has extended the methods of this paper to compute the ECH of (most) T^2-bundles over S^1. Some more recent applications of this paper are described in this blog post.

• [4] Gluing pseudoholomorphic curves along branched covered cylinders I (with C. H. Taubes.)
Journal of Symplectic Geometry 5 (2007), 43-137. ps
This paper and its sequel [5] contain the gluing analysis to prove that d^2=0 in embedded contact homology and periodic Floer homology. This requires an extension of the standard gluing machinery, in order to glue together two J-holomorphic curves in R x Y whose ends are at the same Reeb orbits, but with different covering multiplicities. To glue two such curves together, we need to insert a branched cover of an R-invariant cylinder between them. The number of gluings is given by a count of zeroes of a certain section of an obstruction bundle over a noncompact moduli space of branched covered cylinders. We obtain a general combinatorial formula for the number of gluings as a certain sum over labeled forests. In the case of interest for ECH and PFH, the number of gluings comes out to be 1. This is just what is needed to show that d^2 counts the number of boundary points of a compact 1-manifold, so that d^2=0. Paper [4] explains the more algebraic aspects of this story, using an analytic result from [5].

• [5] Gluing pseudoholomorphic curves along branched covered cylinders II (with C. H. Taubes.)
Journal of Symplectic Geometry 7 (2009), 29-133. pdf
This paper completes all the analysis that was needed in [4]. The gluing technique explained here is in principle applicable to more gluing problems. We also prove some lemmas concerning the generic behavior of pseudoholomorphic curves in symplectizations, which may be of independent interest.

• [6] The embedded contact homology index revisited
New perspectives and challenges in symplectic field theory ("Yashafest"), 263-297, CRM Proc. Lecture Notes, 49. AMS, 2009, nearly final version available at arXiv:0805.1240. Erratum.
In this paper we refine the relative grading on ECH to an absolute grading, which associates to each union of Reeb orbits a homotopy class of oriented 2-plane fields. This is obtained by modifying the contact plane field in a canonical (up to homotopy) manner in a neighborhood of each Reeb orbit in the union. We also simplify the proof of the ECH index inequality, and extend it to symplectic cobordisms between three-manifolds with Hamiltonian structures. Included are new inequalities on the ECH index of unions and multiple covers of holomorphic curves in cobordisms. Finally, we define a new relative filtration on ECH, or any other kind of contact homology of a contact 3-manifold, which is similar to the ECH index and related to the Euler characteristic of holomorphic curves. This does not give new topological invariants except possibly in special situations, but it is a useful computational tool.

• [7] The Weinstein conjecture for stable Hamiltonian structures (with C. H. Taubes.)
We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3-manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y. We prove that if Y is not a T^2-bundle over S^1, then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3-manifold with a contact form such that all Reeb orbits are nondegenerate and ellitpic, then Y is a lens space. Related arguments show that if Y is a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate, and if Y is not a lens space, then there exist at least three distinct embedded Reeb orbits.

• [8] Embedded contact homology and its applications
Proceedings of the 2010 ICM, vol. II, 1022-1041, nearly final version available at arXiv:1003.3209.
In this article we first give an overview of the definition of embedded contact homology. We then outline its applications to generalizations of the Weinstein conjecture, the Arnold chord conjecture, and obstructions to symplectic embeddings in four dimensions. Here are the slides from the associated talk.

• [9] Sutures and contact homology I (with V. Colin, P. Ghiggini, and K. Honda)
Among other things, this paper defines a version of ECH for sutured 3-manifolds, which is conjecturally isomorphic to the sutured (Heegaard) Floer homology of Juhasz.

• [10] Proof of the Arnold chord conjecture in three dimensions I (with C. H. Taubes)
arXiv:1004.4319, Math. Res. Lett. 18 (2011), 295-313.
We prove that every Legendrian knot in a closed three-manifold with a contact form has a Reeb chord. The proof uses another theorem, asserting that an exact symplectic cobordism between contact 3-manifolds induces a map on (filtered) embedded contact homology satisfying certain axioms. The latter theorem is proved in [12] using Seiberg-Witten theory.

• [11] Quantitative embedded contact homology
J. Differential Geometry 88 (2011) 231-266, close to final version at arXiv:1005.2260 (Conjecture 8.5 in the arxiv version is false and does not appear in the published version)
Define a "Liouville domain" to be a compact exact symplectic manifold with contact-type boundary. We use embedded contact homology to assign to each four-dimensional Liouville domain (or subset thereof) a sequence of real numbers, which we call "ECH capacities". The ECH capacities of a Liouville domain are defined in terms of the "ECH spectrum" of its boundary, which measures the amount of symplectic action needed to represent certain classes in embedded contact homology. Using cobordism maps on embedded contact homology (defined in joint work with Taubes), we show that the ECH capacities are monotone with respect to symplectic embeddings. We compute the ECH capacities of ellipsoids, polydisks, certain subsets of the cotangent bundle of T2, and disjoint unions of examples for which the ECH capacities are known. The resulting symplectic embedding obstructions are sharp in some interesting cases, for example for the problem of embedding an ellipsoid into a ball (as shown by work of McDuff-Schlenk) or embedding a disjoint union of balls into a ball. We also state and present evidence for a conjecture under which the asymptotics of the ECH capacities of a Liouville domain recover its symplectic volume. There is some followup discussion in this blog post.

• [12] Proof of the Arnold chord conjecture in three dimensions II (with C. H. Taubes)
We complete the proof of the three-dimensional chord conjecture from [10] by showing that an exact symplectic cobordism between contact three-manifolds induces a map on (filtered) embedded contact homology satisfying certain axioms. We also prove that filtered embedded contact homology does not depend on the choice of almost complex structure used to define it. For more discussion, see this blog post.

• [13] From one Reeb orbit to two (with D. Cristofaro-Gardiner)
J. Diff. Geom. 102 (2016), 25--36. arXiv version, blog post.

• [14] The asymptotics of ECH capacities (with D. Cristofaro-Gardiner and V.G.B. Ramos)
Invent. Math. 199 (2015), 187-214, arXiv:1210.2167. blog post

• [15] Lecture notes on embedded contact homology
Contact and symplectic topology, Bolyai Soc. Math. Stud. 26 (2014), 389-484. Nearly final version at arXiv:1303.5789. blog post

• [16] Symplectic embeddings into four-dimensional concave toric domains (with K. Choi, D. Cristofaro-Gardiner, D. Frenkel and V.G.B. Ramos)
Journal of Topology 7 (2014), 1054-1076. arXiv:1310.6647, blog post.

• [17] Beyond ECH capacities
Geometry and Topology 20 (2016), 1085-1126. Almost final version at arXiv:1409.1352. blog post.

• [18] Mean action and the Calabi invariant
Journal of Modern Dynamics 10 (2016), 511-539. Almost final version at arXiv:1509.02183. blog post.

• [19] Torsion contact forms in three dimensions have two or infinitely many Reeb orbits (with D. Cristofaro-Gardiner and D. Pomerleano)
Geometry and Topology 23 (2019), 3601-3645. Almost final version at arXiv:1701.02262.

• [20] ECH capacities and the Ruelle invariant
arXiv:1910.08260.