Computations of complex equivariant bordism rings

We give explicit computations of the coefficients of homotopical complex equivariant cobordism theory MU^G, when G is abelian. We present a set of generators which is complete for any abelian group. We present a set of relations which is complete when G is cyclic and which we conjecture to be complete in general. We proceed by first computing the localization of MU^G obtained by inverting Euler classes of representations. We then define a family of operations which essentially divide by Euler classes and use these operations to define our generating sets. We give geometric applications of these computations to the study of equivariant genera, circle actions on four-manifolds, and cobordism relations between Lens spaces.