Automatic transversality in contact homology I: Regularity

TL;DR

This paper establishes automatic transversality for a large class of dynamically separated contact forms in dimension 3, enabling well-defined cylindrical contact homology and advancing the understanding of contact invariants.

Contribution

It introduces the class of dynamically separated contact forms and proves automatic transversality, facilitating the construction of contact homology in these cases.

Findings

Automatic transversality holds for dynamically separated contact forms.

Dynamically separated forms satisfy a uniform growth condition on Conley-Zehnder index.

This framework allows defining a chain complex for cylindrical contact homology.

Abstract

This paper helps to clarify the status of cylindrical contact homology, a conjectured contact invariant introduced by Eliashberg, Givental, and Hofer in 2000. We explain how heuristic arguments fail to yield a well-defined homological invariant in the presence of multiply covered curves. We then introduce a large subclass of dynamically convex contact forms in dimension 3, termed dynamically separated, and demonstrate automatic transversality holds, therby allowing us to define the desired chain complex. The Reeb orbits of dynamically separated contact forms satisfy a uniform growth condition on their Conley-Zehnder index under iteration, typically up to large action; see Definition 1.15 These contact forms arise naturally as perturbations of Morse-Bott contact forms such as those associated to $S^1$-bundles. In subsequent work, we give a direct proof of invariance for this subclass and, when further proportionality holds between the index and action, powerful geometric computations in a wide variety of examples.