Minimal hyperspheres of arbitrarily large Morse index

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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 S⁴ 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.