Automatic transversality in contact homology II: filtrations and computations

Abstract

This paper is the sequel to the previous paper [Nelson, Abh. Math. Semin. Univ. Hambg. 85 (2015) 125–179], 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. 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 [Symplectic field theory and its applications, International Congress of Mathematicians I (European Mathematical Society, Zürich, 2007) 217–246] in dimension three.