2 00 1 Curvature , Covering Spaces , and Seiberg - Witten Theory

We point out that there are compact 4-manifolds which do not admit metrics of positive scalar curvature, but nonetheless have finite covering spaces which do carry such metrics. Moreover, passing from a 4-manifold to a covering space sometimes actually changes the sign of the Yamabe invariant. As was first pointed out by Bérard Bergery [1], there exist, in dimensions ≡ 1 or 2 mod 8, n ≥ 9, certain smooth compact manifolds M which don’t admit metrics of positive scalar curvature, but which nonetheless have finite coverings that do admit such metrics. For example, let Σ be an exotic 9sphere which does not bound a spin manifold, and consider the connected sum M = (S × RP)#Σ. On one hand, M is a spin manifold with nonzero Hitchin invariant â(M) ∈ Z2, so [5] there are harmonic spinors on M for every choice of metric; the Lichnerowicz Weitzenböck formula therefore tells us that no metric on M can have positive scalar curvature. On the other hand, the universal cover M̃ = (S2×S7)#2Σ of M is diffeomorphic to S × S, and so admits positive scalar metrics — e.g. the standard product metric. The purpose of this note is to point out that the same phenomenon occurs in dimension 4. Moreover, passing from a 4-manifold to one of its covering spaces can even change the sign of the Yamabe invariant. Recall that Yamabe ∗Supported in part by NSF grant DMS-0072591.