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 $\Lambda_n$, depending only on the dimension $n$ of $M$, such that $$ \sigma(N) \geq \min{\sigma(M),\Lambda_n}. $$