  A strategy for constructing an embedded sphere in a 4-manifold realizing a
given homology class which has been successfully applied in the past is to
represent the class as a first step stably by an embedded sphere, i.e. after
adding products of 2-spheres, and to move that sphere back into the original
manifold. In this paper, we study under what conditions the first step of this
approach can be carried out if the 4-manifold at hand is not simply connected.
One of our main results is that there are - apart from the well known Arf
invariant - additional bordism theoretical obstructions to stably representing
homology classes by embedded spheres.
