Contact homology and virtual fundamental cycles

Abstract

We give a construction of contact homology in the sense of Eliashberg-Givental-Hofer. Specifically, we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo-holomorphic curves.The aim of this work is to provide a rigorous construction of contact homology, an invariant of contact manifolds and symplectic cobordisms due to Eliashberg-Givental-Hofer [Eli98, EGH00]. The contact homology of a contact manifold (Y, ξ) is defined by counting pseudo-holomorphic curves in the sense of Gromov [Gro85] in its symplectization R × Y . The main problem we solve in this paper is simply to give a rigorous definition of these curve counts, the essential difficulty being that the moduli spaces of such curves are usually not cut out transversally. It is therefore necessary to construct the virtual fundamental cycles of these moduli spaces (which play the same enumerative role that the ordinary fundamental cycles do for transversally cut out moduli spaces). For this construction, we use the framework developed in [Par16]. Our methods are quite general, and apply equally well to many other moduli spaces of interest.We use a compactification of the relevant moduli spaces which is smaller than the compactification considered in [EGH00, BEHWZ03]. Roughly speaking, for curves in symplectizations R × Y , we do not keep track of the relative vertical positions of different components (in particular, no trivial cylinders appear). Our compactification is more convenient for proving the master equations of contact homology: the codimension one boundary strata in our compactification correspond bijectively with the desired terms in the "master equations", whereas the compactification from [EGH00, BEHWZ03] contains additional codimension one boundary strata. If we were to use the compactification from [EGH00, BEHWZ03], we would need to additionally argue that the contribution of each such extra codimension one boundary stratum vanishes.Remark 0.1 (Historical discussion). The theory of pseudo-holomorphic curves in closed symplectic manifolds was founded by Gromov [Gro85]. Hofer's breakthrough work on the threedimensional Weinstein conjecture [Hof93] introduced pseudo-holomorphic curves in symplectizations and their relationship with Reeb dynamics. The analytic theory of such curves was then further developed by Hofer-Wysocki-Zehnder [HWZ96, HWZ98, HWZ95, HWZ99, *

Definition of cylindrical contact homology in dimension three

“…One of the usual properties of abstract perturbation theory is that when perturbations are not needed (that is, the relevant moduli spaces are already transversely cut out) the counts of holomorphic curves without perturbations agrees with the counts with perturbations. Hence the definition of cylindrical contact homology in this paper agrees with those of [1,19].…”

“…Corollary 8. 19. For sufficiently small δ > 0, sufficiently large j > 0, and sufficiently large R > 0,…”

Multiplicity of closed Reeb orbits on prequantization bundles

“…The foundational aspects of the contact homology theory are still to be fully laid down. We refer the reader to [28,29,30] for the polyfold approach to this theory and to [36,37] for the virtual cycle approach and further references.…”

Topological entropy for Reeb vector fields in dimension three via open book decompositions

“…Hypothesis H has now been proven by Bao-Honda [BH]; Pardon [Pa1,Pa2] gives an algebraic substitute. Establishing Hypothesis H could also be done using the polyfold technology of Hofer-Wysocki-Zehnder [HWZ2] or the Kuranishi structures of Fukaya-Oh-Ohta-Ono [FO3].…”