Legendrian ECH

I will explain a construction of a Legendrian version of embedded contact homology (ECH) for a sutured contact manifold Y along with a collection of Legendrians L contained in the boundary. The chain complex is generated by sets of Reeb orbits and Reeb chords, and the differential counts certain embedded curves of "relative ECH index" 1. This version of ECH and it's PFH analogue will help to categorify the zeta function of gradient flows of circle valued Morse functions discussed by Hutchings in his thesis. It also provides a unification between the cylindrical formulation of Heegaard-Floer theory given by Lipschitz and standard ECH.

This is joint work in preparation with Oliver Edtmair, Luya Wang, Yuan Yao and Ziwen Zhao.