Investigation of J-holomorphic curves in M3 x S1

In this thesis, we examine the moduli space of pseudoholomorphic curves embedded in a four-dimensional symplectic manifold of the form X = M 3 × S 1 , where M fibres over the circle with fibre F having genus g . A pseudoholomorphic structure J on such a manifold X will be invariant in the S 1 direction. For generic choice of S 1 -invariant J , the moduli space of J -holomorphic curves of class E = F + gT is smooth. In fact, the moduli space will be a torus or collection of tori. We also explore one component of the moduli space for a particular M 3 = F × [0,1]/[straight phi] where [straight phi] g has a fixed point.