Positive scalar curvature and exotic structures on simply connected four manifolds

TL;DR

This paper investigates the validity of Gromov's band width inequality and Rosenberg's $S^1$-stability conjecture for simply connected four-manifolds, showing they hold up to homeomorphism despite known counterexamples in the smooth category.

Contribution

It demonstrates that these geometric inequalities and conjectures are true for simply connected four-manifolds when considering topological equivalence rather than smooth structures.

Findings

Gromov's band width inequality holds up to homeomorphism in simply connected four-manifolds.

Rosenberg's $S^1$-stability conjecture is valid up to homeomorphism in these manifolds.

Results extend to certain non-simply connected four-manifolds.

Abstract

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.