Algebraic realization problems for low dimensional G manifolds

In this paper we prove that every real G vector bundles over G circles or on effective G surfaces can be realized by strongly algebraic G vector bundles for finite Abelian groups G. Using this result we prove that every closed orientable smooth three dimensional G manifold is G diffeomorphic to a nonsingular real algebraic G variety for any finite Abelian group G. We also prove that for any finite group G the algebraic realization of smooth G vector bundles over effective G surfaces can be reduced to the algebraic realization of smooth G vector bundles over G circles.