  The stability of Yang-Mills bundles over the usual $S^4$ space-time manifold
is investigated according to the topological methods. The necessary gauge- and
topological invaraint criterion for the exsitence of the related critical
points is defined. It is shown that according to this criterion there exists no
critical point even for the action functional of the standard U(1) gauge theory
of electrodynamics on a $S^4$ manifold in view of its topological structure and
therefore such a theory can not be stable. We will discuss also a general
consequence of this result according to which for a stable U(1) Yang-Mills
theory over a compact 4-manifold, this manifold should possess some self
consistent compact 2-manifold substructure. These results are also in agreement
with the known very general result for the {\it structural stability} of
dynamical systems.
