  The aim of the present paper is to investigate the behavior of the spectrum
of the Neumann Laplacian in domains with little holes excised from the
interior. More precisely, we consider the eigenvalues of the Laplacian with
homogeneous Neumann boundary conditions on a bounded, Lipschitz domain. Then,
we singularly perturb the domain by removing Lipschitz sets which are "small"
in a suitable sense and satisfy a uniform extension property. In this context,
we provide an asymptotic expansion for all the eigenvalues of the perturbed
problem which are converging to simple eigenvalues of the limit one. The
eigenvalue variation turns out to depend on a geometric quantity resembling the
notion of (boundary) torsional rigidity: understanding this fact is one of the
main contributions of the present paper. In the particular case of a hole
shrinking to a point, through a fine blow-up analysis, we identify the exact
vanishing order of such a quantity and we establish some connections between
the location of the hole and the sign of the eigenvalue variation.
