A surgery formula for the second Yamabe invariant

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\ge 3$. For a metric $g$ on $M$, we let ${\text{la}}_2(g)$ be the second eigenvalue of the Yamabe operator $L_g:=\frac{4(n-1)}{n-2}{\mathrm{\Delta }}_g+{\text{scal}}_g$. Then, the second Yamabe invariant is defined as $${\text{si}}_2(M)\text{definedas}\sup \underset{h\in [g]}{\inf }{\text{la}}_2(h)\text{Vol}(M,h{)}^{2\mathrm{/}n}\mathrm{.}$$ where the supremum is taken over all metrics $g$ and the infimum is taken over the metrics in the conformal class $[g]$. Assume that ${\text{si}}_2(M)>0$. In the spirit of \cite{ammann.dahl.humbert:08}, we prove that if $N$ is obtained from $M$ by a $k$-dimensional surgery ($0\le k\le n-3$), there exists a positive constant ${\mathrm{\Lambda }}_n$ depending only on $n$ such that ${\text{si}}_2(N)\ge \min ({\sigma }_2(M),{\mathrm{\Lambda }}_n)$. We then give some topological conclusions of this result.