  Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. For a
metric $g$ on $M$, we let $\la_2(g)$ be the second eigenvalue of the Yamabe
operator $L_g:= \frac{4(n-1)}{n-2} \Delta_g + \scal_g$. Then, the second Yamabe
invariant is defined as $$ \si_2(M) \definedas \sup \inf_{h \in [g]} \la_2(h)
\Vol(M,h)^{2/n}. $$ where the supremum is taken over all metrics $g$ and the
infimum is taken over the metrics in the conformal class $[g]$. Assume that
$\si_2(M)>0$. In the spirit of \cite{ammann.dahl.humbert:08}, we prove that if
$N$ is obtained from $M$ by a $k$-dimensional surgery ($0 \leq k \leq n-3$),
there exists a positive constant $\Lambda_n$ depending only on $n$ such that
$\si_2(N) \geq \min(\sigma_2(M), \Lambda_n)$. We then give some topological
conclusions of this result.
