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

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