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We define a new differential invariant a compact manifold by $V_{\mathcalM}(M)=\inf_g V_c(M,[g])$, where $V_c(M,[g])$ is the conformal volume of $M$ forthe conformal class $[g]$, and prove that it is uniformly bounded above. Themain motivation is that this bound provides a upper bound of theFriedlander-Nadirashvili invariant defined by $\inf_g\sup_{\tildeg\in[g]}\lambda_1(M,\tilde g)\Vol(M,\tilde g)^{\frac 2n}$.  The proof relies on the study of the behaviour of $V_{\mathcal M}(M)$ whenone performs surgeries on $M$.

Premi\`ere valeur propre du laplacien, volume conforme et chirurgies