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Fabert, Oliver (2008): Transversality Results and Computations in Symplectic Field Theory. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
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Abstract

Although the definition of symplectic field theory suggests that one has to count holomorphic curves in cylindrical manifolds equipped with a cylindrical almost complex structure, it is already well-known from Gromov-Witten theory that, due to the presence of multiply-covered curves, we in general cannot achieve transversality for all moduli spaces even for generic choices. In this thesis we treat the transversality problem of symplectic field theory in two important cases. In the first part of this thesis we are concerned with the rational symplectic field theory of Hamiltonian mapping tori, which is also called the Floer case. For this observe that in the general geometric setup for symplectic field theory, the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is given by the Floer homologies of powers of the symplectomorphism, the other algebraic invariants of symplectic field theory provide natural generalizations of symplectic Floer homology. For symplectically aspherical manifolds and Hamiltonian symplectomorphisms we study the moduli spaces of rational curves and prove a transversality result, which does not need the polyfold theory by Hofer, Wysocki and Zehnder and allows us to compute the full contact homology. The second part of this thesis is devoted to the branched covers of trivial cylinders over closed Reeb orbits, which are the trivial examples of punctured holomorphic curves studied in rational symplectic field theory. Since all moduli spaces of trivial curves with virtual dimension one cannot be regular, we use obstruction bundles in order to find compact perturbations making the Cauchy-Riemann operator transversal to the zero section and show that the algebraic count of elements in the resulting regular moduli spaces is zero. Once the analytical foundations of symplectic field theory are established, our result implies that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration. After introducing additional marked points and differential forms on the target manifold we finally use our result to compute the second page of the corresponding spectral sequence for filtered complexes.