Let D be a surface with a symplectic form, let J be a symplectomorphism of D, and let Y be the mapping torus of J. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in Â2Y, with cylindrical ends asymptotic to periodic orbits of J 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.
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Michael Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations. J. Eur. Math. Soc. 4 (2002), no. 4, pp. 313–361DOI 10.1007/S100970100041