In dimensions other than four, positive scalar curvature imposes strong constraints on manifold topology. Gromov's band width inequality bounds the width of a band (product manifold) that admits positive scalar curvature. Rosenberg's S¹-stability conjecture predicts when a manifold times a circle admits positive scalar curvature. Both are proven in dimensions ≥ 5.
In dimension four, both fail. The failure is specifically smooth: Seiberg-Witten invariants detect exotic smooth structures — manifolds that are homeomorphic (topologically the same) but not diffeomorphic (smoothly different). These exotic structures provide counterexamples. A four-manifold may admit positive scalar curvature with one smooth structure but not another, despite being topologically identical.
The paper shows that the failure is entirely a smooth phenomenon. Relaxing diffeomorphism to homeomorphism — asking “up to topological equivalence” rather than “up to smooth equivalence” — restores both conjectures for simply connected four-manifolds. The constraints that positive scalar curvature imposes on topology are the same in dimension four as in higher dimensions. The anomaly is in the smooth category, not the topological category.
The structural point: dimension four's pathologies are properties of the smooth structure, not of the space itself. The same topological manifold can satisfy or violate curvature constraints depending on which smooth structure it carries. The topology obeys the expected rules; the smooth structure breaks them. The rescue is not a weaker result — it is a diagnosis: the problem lives in the gap between topology and smoothness, which is uniquely wide in dimension four.