Automatic transversality in contact homology II: filtrations and computations

TL;DR

This paper develops a filtration approach for cylindrical contact homology in dimension three, proving invariance and independence results for dynamically separated contact forms and confirming a conjecture for prequantization bundles.

Contribution

It introduces a filtration by action on the chain complex and demonstrates how to compute cylindrical contact homology via direct limits for dynamically separated contact forms.

Findings

Established a filtration method for cylindrical contact homology

Proved invariance of the filtered cylindrical contact homology

Confirmed a conjecture relating to prequantization bundles

Abstract

This paper is the sequel to the previous paper [Ne15], which showed that sufficient regularity exists to define cylindrical contact homology in dimension three for nondegenerate dynamically separated contact forms, a subclass of dynamically convex contact forms. The Reeb orbits of these so-called dynamically separated contact forms satisfy a uniform growth condition on their Conley-Zehnder indices with respect to a free homotopy class; see Definition 1.7. {Given a contact form which is dynamically separated up to large action, we demonstrate a filtration by action on the chain complex and show how to obtain the desired cylindrical contact homology by taking direct limits.} We give a direct proof of invariance of cylindrical contact homology within the class of dynamically separated contact forms, {and elucidate the independence of the filtered cylindrical contact homology with respect to the choice of the dynamically separated contact form and almost complex structure.} We also show that these regularity results are compatible with geometric methods of computing cylindrical contact homology of prequantization bundles, proving a conjecture of Eliashberg [El07] in dimension three.