CONTACT HOMOLOGY AND VIRTUAL FUNDAMENTAL CYCLES

Abstract

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 [22, 23]. The contact homology of a contact manifold (Y,ξ) is defined by counting pseudo-holomorphic curves in the sense of Gromov [42] 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 there-fore 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 [83]. Our methods are quite general, and apply equally well to many other moduli spaces of interest.