2023/02/14 by Chaidez, Julian, Edtmair, Oliver, Wang, Luya +2
#53D10 #53D40 #57R58 #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2302.07259
We give a construction of embedded contact homology (ECH) for a contact 3-manifold Y with convex sutured boundary and a pair of Legendrians Λ+ and Λ- contained in ∂ Y satisfying an exactness condition. The chain complex is generated by certain configurations of closed Reeb orbits of Y and Reeb chords of Λ+ to Λ-. The main ingredients include: a general Legendrian adjunction formula for curves in ℝ × Y with boundary on ℝ × Λ; 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,Λ) of a closed contact 3-manifold Y and a Legendrian link Λ can also be defined using this machinery after passing to a sutured link complement. This work builds on ideas present in Colin-Ghiggini-Honda's proof of the equivalence of Heegaard-Floer 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.