Recent work by S. Donaldson yields an invariant of the diffeomorphism type of four-dimensional manifolds. In the case of an algebraic surface this invariant can be expressed in terms of the fundamental class of the surface and the moduli space of stable SU(2) bundles V on the surface whose Chern classes are c(,1)(V) = 0 and c(,2)(V) = 1. R. Friedman and J. Morgan have calculated this invariant in the case of certain simply-connected elliptic surfaces. In this dissertation we study elliptic surfaces X with cyclic fundamental group, dim H('2)(X,(//R)) = 10 and dim H(,+)('2)(X,(//R)) = 1. We show that for every finite cyclic group G there is a topological four-dimensional manifold with fundamental group G that allows infinitely many different diffeomorphism types