Scalar curvature, covering spaces, and Seiberg-Witten theory

The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalarcurvature Riemannian metrics g on M . (To be precise, one only considers those constant-scalar-curvature metrics which are Yamabe minimizers, but this technicality does not, e.g., affect the sign of the answer.) In this article, it is shown that many 4-manifolds M with Y(M) < 0 have have finite covering spaces M̃ with Y(M̃) > 0. Two decades ago, Lionel Bérard Bergery [2] pointed out that there are highdimensional smooth compact manifolds M which do not admit metrics of positive scalar curvature, but which nevertheless have finite coverings that do admit such metrics. For example, let Σ be an exotic 9-sphere which does not bound a spin manifold, and consider the connected sum M = (S2×RP)#Σ. On one hand, M is a spin manifold with nonzero Hitchin invariant â(M) ∈ Z2, so [7] there are harmonic spinors on M for every choice of metric; the Lichnerowicz Weitzenböck formula for the Dirac operator therefore tells us that no metric on M can have positive scalar curvature. On the other hand, the universal cover M̃ = (S × S)#2Σ of M is diffeomorphic to S × S, on which the obvious product metric certainly has positive scalar curvature. As will be shown here, the same phenomenon also occurs in dimension four. Indeed, far more is true. In the process of passing from a 4-manifold to a finite cover, it is even possible to change the sign of the Yamabe invariant. The Yamabe invariant is a diffeomorphism invariant that historically arose from an attempt to construct Einstein metrics (metrics of constant Ricci curvature) on smooth compact manifolds. A standard computation [3] shows that the Einstein metrics on any given smooth compact manifold M of dimension n > 2 are exactly the critical points of the normalized total scalar curvature S(g) = V (2−n)/n g ∫