The classical Dirichlet-to-Neumann (DN) operator transforms functions defined on the boundary of a compact Riemannian manifold with boundary and naturally arises in the problem of electrical impedance tomography. In a recent paper, Belishev and Sharafutdinov generalized the definition of the DN map to differential forms and showed that this operator completely determines the additive cohomology structure and the long exact sequence of the manifold with boundary. In this talk I will give an overview of Belishev and Sharafutdinov's results as well as some hints of a connection to some new invariants of Riemannian manifolds with boundary arising from a refinement of the Hodge-Morrey-Friedrichs decomposition.