Stokes’ Theorem on Manifolds is often express as follows:
Given a differential m-form $\omega$ whose support is the $C^{\infty}$ $m$-dimensional compact manifold ${\cal{M}}$ with boundary $\partial{\cal{M}}$ then \begin{equation} \int_{\cal{M}}d\omega=\int_{\partial\cal{M}}\omega. \end{equation} Also Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology.
My questions are:
What is the lowest regularity on a Sobolev Space or similar functional space we can assume on $\omega$?
Is there something similar to De-Rham cohomology in the low regularity setting?