Reconstruction of Betti numbers of manifolds for anisotropic Maxwell and Dirac systems

We consider an invariant formulation of the system of Maxwell’s equations for an anisotropic medium on a compact orientable Riemannian 3-manifold $(M,g)$ with nonempty boundary. The system can be completed to a Dirac type first order system on the manifold. We show that the Betti numbers of the manifold can be recovered from the dynamical response operator for the Dirac system given on a part of the boundary. In the case of the original physical Maxwell system, assuming that the entire boundary is known, all Betti numbers of the manifold can also be determined from the dynamical response operator given on a part of the boundary. Physically, this operator maps the tangential component of the electric field into the tangential component of the magnetic field on the boundary.