S 1 $S^1$ -equivariant contact homology for hypertight contact forms

Abstract

In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three-manifold with a dynamically convex contact form. However, we did not show that this cylindrical contact homology is an invariant of the contact structure. In the present paper, we define ‘nonequivariant contact homology’ and ‘$S^1$-equivariant contact homology’, both with integer coefficients, for a contact form on a closed manifold in any dimension with no contractible Reeb orbits. We prove that these contact homologies depend only on the contact structure. Our construction uses Morse–Bott theory and is related to the positive $S^1$-equivariant symplectic homology of Bourgeois-Oancea. However, instead of working with Hamiltonian Floer homology, we work directly in contact geometry, using families of almost complex structures. When cylindrical contact homology can also be defined, it agrees with the tensor product of the $S^1$-equivariant contact homology with $Q$. We also present examples showing that the $S^1$-equivariant contact homology contains interesting torsion information. In a subsequent paper, we will use obstruction bundle gluing to extend the above story to closed three-manifolds with dynamically convex contact forms, which in particular will prove that their cylindrical contact homology has a lift to integer coefficients which depends only on the contact structure.