Geometric transversality in higher genus Gromov-Witten theory

The construction of manifold structures and fundamental classes on the (compactified) moduli spaces appearing in Gromov-Witten theory is a long-standing problem. Up until recently, most successful approaches involved the imposition of topological constraints like semi-positivity on the underlying symplectic manifold to deal with this situation. One conceptually very appealing approach that removed most of these restrictions is the approach by K. Cieliebak and K. Mohnke via complex hypersurfaces, [CM07]. In contrast to other approaches using abstract perturbation theory, it has the advantage that the objects to be studied still are spaces of holomorphic maps defined on Riemann surfaces. This article aims to generalise this from the case of surfaces of genus 0 dealt with in [CM07] to the general case, also using some of the methods from [IP03] and symplectic field theory, namely the compactness results from [BEH+03].