Automatic transversality in contact homology I: Regularity

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.

arXiv comment68 pages, added more information about bad Reeb orbits, added a proof of a beloved folk theorem concerning the factorization of multiply covered curves, contains expository revisions helpfully suggested by the referee

References in corpus (11)

• An exact sequence for contact- and symplectic homology
• Relative asymptotic behavior of pseudoholomorphic half-cylinders
• Technical details on Kuranishi structure and virtual fundamental chain
• $S^1$-equivariant symplectic homology and linearized contact homology
• Lectures on Holomorphic Curves in Symplectic and Contact Geometry
• Virtual neighborhood technique for pseudo-holomorphic spheres
• A natural Gromov-Witten virtual fundamental class
• On Symplectic Sum Formulas in Gromov-Witten Theory
• Contact Homology of Orbit Complements and Implied Existence
• Contact homology of $S^1$-bundles over some symplectically reduced orbifolds
• Cylindrical contact homology for dynamically convex contact forms in three dimensions

Cited by in corpus (1)

• Contact non-squeezing at large scale in ${\mathbb{R}}^{2n}\times S^1$