Transversality Results and Computations in Symplectic Field Theory

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.