  We show that the Morse index of a closed minimal hypersurface in a
four-dimensional Riemannian manifold cannot be bound in terms of the volume and
the topological invariants of the hypersurface itself by presenting a method
for constructing Riemannian metrics on S^4 that admit embedded minimal
hyperspheres of uniformly bounded volume and arbitrarily large Morse index. The
phenomena we exhibit are in striking contrast with the three-dimensional
compactness results by Choi-Schoen.
