Smooth Yamabe invariant and surgery

We prove a surgery formula for the smooth Yamabe invariant $\sigma (M)$ of a compact manifold $M$. Assume that $N$ is obtained from $M$ by surgery of codimension at least 3. We prove the existence of a positive number ${\mathrm{\Lambda }}_n$, depending only on the dimension $n$ of $M$, such that $$\sigma (N)\ge \min \sigma (M),{\mathrm{\Lambda }}_n\mathrm{.}$$