An index inequality for embedded pseudoholomorphic curves in symplectizations

Abstract

Let $Σ$ be a surface with a symplectic form, let $ϕ$ be a symplectomorphism of $Σ$, and let $Y$ be the mapping torus of $ϕ$. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in $\R\times Y$, with cylindrical ends asymptotic to periodic orbits of $ϕ$ 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 homology.