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 \(\phi\). This paper studies the moduli spaces of embedded pseudoholomorphic curves in \(\mathbb{R} \times Y\), with cylindrical ends asymptotic to periodic orbits of \(\phi\) or of multiple covers of such periodic orbits. It is shown that the dimension of these moduli spaces are bounded above by an index defined for every relative homology class. The details of the calculation of the virtual dimension are delicate numerical arguments involving the invariants coming from the Poincaré return map for each periodic orbit. This is applied to deduce compactness results for these moduli spaces, which provide one step in the direction of defining periodic Floer homology. This work establishes some of the foundations for a program of the author with Michael Thaddeus, aimed to understand the Seiberg-Witten-Floer homology of \(Y\) in terms of pseudo-holomorphic curves of \(\mathbb{R} \times Y\). An analogue of periodic Floer homology for a contact three-manifold would give an interesting variant of the symplectic field theory of [\textit{Y. Eliashberg, A. Givental} and \textit{H. Hofer}, Introduction to symplectic field theory, GAFA 2001, Tel Aviv, Israel, 1999, Part II, 560-673 (2000; Zbl 0989.81114)].
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