An index inequality for embedded pseudoholomorphic curves in symplectizations

Details An index inequality for embedded pseudoholomorphic curves in symplectizations An index inequality for embedded pseudoholomorphic curves in symplectizations

2001-12-17

arxiv.org/abs/math/0112165v1

Mathematics

Symplectic Geometry

60 pages, LaTeX 2e

Scientific paper

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 $\R\times Y$, 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 homology.

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