Positive scalar curvature, diffeomorphisms and the Seiberg–Witten invariants

Abstract

We study the space of positive scalar curvature (psc) metrics on a 4–manifold, and give examples of simply connected manifolds for which it is disconnected. These examples imply that concordance of psc metrics does not imply isotopy of such metrics. This is demonstrated using a modification of the 1–parameter Seiberg–Witten invariants which we introduced in earlier work. The invariant shows that the diffeomorphism group of the underlying 4–manifold is disconnected. We also study the moduli space of positive scalar curvature metrics modulo diffeomorphism, and give examples to show that this space can be disconnected. The (non-orientable) 4–manifolds in this case are explicitly described, and the components in the moduli space are distinguished by a ${Pin}^c$ eta invariant.