Lecture Notes on Embedded Contact Homology

By Michael Hutchings. 2013

138 citations recorded by the source corpus.

These notes give an introduction to embedded contact homology (ECH) of contact three-manifolds, gathering together many basic notions which are scattered across a number of papers. We also discuss the origins of ECH, including various remarks and examples which have not been previously published. Finally, we review the recent application to four-dimensional symplectic embedding problems. This article is based on lectures given in Budapest and Munich in the summer of 2012, a series of accompanying blog postings at floerhomology.wordpress.com , and related lectures at UC Berkeley in Fall 2012. There is already a brief introduction to ECH in the article of M. Hutchings (in Proceedings of the 2010 ICM, vol. II, pp. 1022–1041, 2010), but the present notes give much more background and detail.

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

View Paper: https://www.semanticscholar.org/paper/1672b8e27bbe9ab0c56e11b8691baa52b8285b86

How researchers see this paper

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

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Symplectic embedding problems, old and new (2017, background): “Excellent surveys on ECH capacities are [63,64], and a quite comprehensive survey on the new results produced by the third revolution is [98].”
The asymptotics of ECH capacities (2012, background): “More details about ECH may be found in [6] and the references therein.”
Torsion contact forms in three dimensions have two or infinitely many Reeb orbits (2017, background): “More details about ECH can be found in the lecture notes [14].”
Symplectic embeddings into four‐dimensional concave toric domains (2013, background): “1 requires some knowledge of the definition of ECH capacities, which we now briefly review; for full details, see [5] or [7].”
Proof of the Arnold chord conjecture in three dimensions, II (2010, unknown): “elow). Although this paper is a sequel to [13], we will not use anything from the latter paper except for some basic deﬁnitions. We begin by brieﬂy reviewing these deﬁnitions. For more about ECH, see [6, 8] and the references therein. 1.1 Embedded contact homology Let Y be a closed oriented 3-manifo”
Beyond ECH capacities (2014, background): “More details can be found in the survey [10].”
Cylindrical contact homology for dynamically convex contact forms in three dimensions (2014, unknown): “If γ is elliptic then θ is an irrational number (due to our assumption that all possibly multiply covered Reeb orbits are nondegenerate), see [18, § 3.2].”
Fivebranes and 3-manifold homology (2016, background): “• via symplectic geometry, “Heegaard Floer homology” HF(M3) [20], • via gauge theory, “Seiberg-Witten Floer homology” or “monopole Floer homology” HM(M3) based on Seiberg-Witten equations [21], • via contact geometry, “embedded contact homology” ECH(M3) [22, 23].”
Symplectic embeddings from concave toric domains into convex ones (2014, background): “rMrowka in [17]. In particular, the homology ECHpY;qdepends neither on the choice of almost complex structure J, nor on , and so we sometimes denote it ECHpYq. For more details about the above, see [16]. Reeb orbits have an action Ap q ‡ which we can extend to a ltration on ECH. Specically, if tp i;”