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# Communications in Analysis and Geometry

## Volume 18 (2010)

### Number 5

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

Pages: 963 – 985

DOI: http://dx.doi.org/10.4310/CAG.2010.v18.n5.a4

#### Authors

#### Abstract

We consider an invariant formulation of the system ofMaxwell's equations for a general anisotropic medium on acompact orientable Riemannian three-manifold $(M,g)$ withnonempty boundary. The system can be completed to a Diractype first-order system on the manifold. We show that theBetti numbers of the manifold can be recovered from thedynamical response operator for the Dirac system given on apart of the boundary. In the case of the original physicalMaxwell system, assuming that the entire boundary is known,all Betti numbers of the manifold can also be determinedfrom the dynamical response operator given on a part of theboundary. Physically, this operator maps the tangentialcomponent of the electric field into the tangentialcomponent of the magnetic field on the boundary.