Computing Embedded Contact Homology in Morse-Bott Settings

TL;DR

This paper develops a method to compute embedded contact homology for Morse-Bott contact forms with Reeb orbits in $S^1$ families, expanding computational techniques in contact topology under certain transversality conditions.

Contribution

It introduces a chain complex for Morse-Bott contact forms using ECH index one cascades and proves it computes the ECH, filling technical gaps in previous literature.

Findings

Defines ECH chain complex via ECH index one cascades.

Proves the chain complex computes the ECH.

Fills foundational gaps for Morse-Bott contact forms.

Abstract

Given a contact three manifold $Y$ with a nondegenerate contact form $\lambda$, and an almost complex structure $J$ compatible with $\lambda$, its embedded contact homology $ECH(Y,\lambda)$ is defined (arXiv:1303.5789) and only depends on the contact structure. In this paper we explain how to compute ECH for Morse-Bott contact forms whose Reeb orbits appear in $S^1$ families, assuming the almost complex structure $J$ can be chosen to satisfy certain transversality conditions (this is the case for instance for boundaries of concave or convex toric domains, or if all the curves of ECH index one have genus zero). We define the ECH chain complex for a Morse-Bott contact form via an enumeration of ECH index one cascades. We prove using gluing results from arXiv:2206.04334 that this chain complex computes the ECH of the contact manifold. This paper and arXiv:2206.04334 fill in some technical foundations for previous calculations in the literature (arXiv:1608.07988, arXiv:math/0410061).