Contact homology of Hamiltonian mapping tori

Abstract

In the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori $M_{\Phi }$ of symplectic manifolds $(M,\omega )$ with symplectomorphisms $\Phi$. While the cylindrical contact homology of $M_{\Phi }$ is given by the Floer homologies of powers of $\Phi$, the other algebraic invariants of symplectic field theory for $M_{\Phi }$ provide natural generalizations of symplectic Floer homology. For symplectically aspherical $M$ and Hamiltonian $\Phi$ we study the moduli spaces of rational curves and prove a transversality result, which does not need the polyfold theory by Hofer, Wysocki and Zehnder. We use our result to compute the full contact homology of $M_{\Phi }\cong S^1\times M$.