  Given a closed orientable hyperbolic manifold of dimension $\neq 3$ we prove
that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on
perpendicular one-forms at zero agrees with the first Betti number of the
manifold. Additionally, we prove that this equality is stable under small
perturbations of the Riemannian metric and simultaneous small perturbations of
the geodesic vector field within the class of contact vector fields. For more
general perturbations we get bounds on the multiplicity of the resonance zero
on all one-forms in terms of the first and zeroth Betti numbers. Furthermore,
we identify for hyperbolic manifolds further resonance spaces whose
multiplicities are given by higher Betti numbers.
