  We study the Yamabe invariant of manifolds obtained as connected sums along
submanifolds of codimension greater than 2. In particular, given a compact
smooth manifold M which does not admit metrics of positive scalar curvature, we
prove that the Yamabe invariant of M is an upper bound for the Yamabe invariant
of any manifold obtained by performing surgery in M on spheres of codimension
greater than 2 .
