  Let $M$ be a compact constant mean curvature surface either in $\mathbb{S}^3$
or $\mathbb{R}^3$. In this paper we prove that the stability index of $M$ is
bounded below by a linear function of the genus. As a by product we obtain a
comparison theorem between the spectrum of the Jacobi operator of $M$ and those
of Hodge Laplacian of $1$-forms on $M$.
