Computations of the Yamabe invariant

For a compact connected manifold $M$ of dimension $n\geq 4$ with no metric of positive scalar curvature, we prove that the Yamabe invariant is unchanged under surgery on spheres of dimension different from 1, $n-2$ and $n-1$. We use this result to give new exact computations of the Yamabe invariant in dimension four and display new examples of compact four-manifolds which do not admit Einstein metrics.

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Published 1 January 1998