An index inequality for embedded pseudoholomorphic curves in symplectizations

Let Sigma be a surface with a symplectic form, let phi be a symplectomorphism of Sigma, and let Y be the mapping torus of phi. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in R x Y, with cylindrical ends asymptotic to periodic orbits of phi or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces. This paper establishes some of the foundations for a program with Michael Thaddeus, to understand the Seiberg-Witten Floer homology of Y in terms of such pseudoholomorphic curves. Analogues of our results should also hold in three dimensional contact topology.