  The Dirichlet-to-Neumann map for differential forms on a Riemannian manifold
with boundary is a generalization of the classical Dirichlet-to-Neumann map
which arises in the problem of Electrical Impedance Tomography. We synthesize
the two different approaches to defining this operator by giving an invariant
definition of the complete Dirichlet-to-Neumann map for differential forms in
terms of two linear operators {\Phi} and {\Psi}. The pair ({\Phi}, {\Psi}) is
equivalent to Joshi and Lionheart's operator {\Pi} and determines Belishev and
Sharafutdinov's operator {\Lambda}. We show that the Betti numbers of the
manifold are determined by {\Phi} and that {\Psi} determines a chain complex
whose homologies are explicitly related to the cohomology groups of the
manifold.
