Positive scalar curvature and exotic structures on simply connected four manifolds

We address Gromov’s band width inequality and Rosenberg’s $S^1$-stability conjecture for simply connected smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants. Nevertheless we show that both of these results hold upon considering simply connected smooth four manifolds up to homeomorphism. We also obtain a related result for non-simply connected smooth four manifolds.