Computations of the Yamabe invariant

For a compact connected manifold M of dimension n greater than 3 and 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 computations of the invariant in dimension four and display new examples of 4-manifolds which do not admit Einstein metrics.