Legendrian embedded contact homology
Abstract
We give a construction of embedded contact homology (ECH) for a contact $3$manifold $Y$ with convex sutured boundary and a pair of Legendrians $\Lambda_+$ and $\Lambda_$ contained in $\partial Y$ satisfying an exactness condition. The chain complex is generated by certain configurations of closed Reeb orbits of $Y$ and Reeb chords of $\Lambda_+$ to $\Lambda_$. The main ingredients include: a general Legendrian adjunction formula for curves in $\mathbb{R} \times Y$ with boundary on $\mathbb{R} \times \Lambda$; a relative writhe bound for curves in contact $3$manifolds asymptotic to Reeb chords; and a Legendrian ECH index with an accompanying ECH index inequality. The (action filtered) Legendrian ECH of any pair $(Y,\Lambda)$ of a closed contact $3$manifold $Y$ and a Legendrian link $\Lambda$ can also be defined using this machinery after passing to a sutured link complement. This work builds on ideas present in ColinGhigginiHonda's proof of the equivalence of HeegaardFloer homology and ECH. The independence of our construction of choices of almost complex structure and contact form should require a new flavor of monopole Floer homology. It is beyond the scope of this paper.
 Publication:

arXiv eprints
 Pub Date:
 February 2023
 DOI:
 10.48550/arXiv.2302.07259
 arXiv:
 arXiv:2302.07259
 Bibcode:
 2023arXiv230207259C
 Keywords:

 Mathematics  Symplectic Geometry;
 53D10;
 57R58;
 53D40
 EPrint:
 78 pages, comments welcome! v2 corrected a few typos in the arXiv submission of v1