  We consider smooth vector bundles over smooth manifolds equipped with
non-smooth geometric data. For nilpotent differential operators acting on these
bundles, we show that the kernels of induced Hodge-Dirac-type operators remain
isomorphic under uniform perturbations of the geometric data. We consider
applications of this to the Hodge-Dirac operator on differential forms induced
by so-called rough Riemannian metrics, which can be of only measurable
coefficient in regularity, on both compact and non-compact settings. As a
consequence, we show that the kernel of the associated non-smooth Hodge-Dirac
operator with respect to a rough Riemannian metric remains isomorphic to smooth
and singular cohomology when the underlying manifold is compact.
