Last December I blogged about the \(S^1\)-stability conjecture for psc-metrics (link). Unfortunately, the result I explained there turned out to be wrong: Counter-examples can be found in dimension 4.
Dimension 4 is special in the theory of psc-metrics since here one can use Seiberg-Witten theory to find obstructions to the existence of psc-metrics on closed manifolds which don’t work in higher dimensions. But dimension 4 is also special since here, due to the failure of the h-cobordism theorem, one encounters frequently(*) manifolds with exotic smooth structures, i.e. manifolds which are homeomorphic but not diffeomorphic to each other. Now interestingly, by a new result by Kumar-Sen (arXiv:2501.01113) these two phenomena are closely related: They proved that, at least in the simply-connected case, the \(S^1\)-stability conjecture is true if we allow ourselves to change the smooth structure.
(*) I would like to say that it happens “more often” than in higher dimensions that a 4-manifold admits exotic structures, but I don’t know if this can be made precise of if it is actually true at all.