The problem of Electrical Impedance Tomography is to determine the conductivity inside a region from knowledge of the voltage-to-current map on the boundary. This problem is equivalent to the problem of determining the Riemannian metric on a manifold with boundary from the Dirichlet-to-Neumann map on the boundary. In this talk I will discuss the "right" way to generalize the Dirichlet-to-Neumann map to an operator on differential forms and show how the Betti numbers of the manifold can easily be recovered from knowledge of this operator. This is joint work with Vladimir Sharafutdinov.