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

TL;DR

This paper demonstrates that the Betti numbers of a 3-manifold can be reconstructed from boundary measurements of Maxwell and Dirac systems in anisotropic media, linking topological invariants to physical response operators.

Contribution

It introduces a method to recover the Betti numbers of a manifold from boundary data of Maxwell and Dirac systems, extending inverse boundary value problem techniques to topological invariants.

Findings

Betti numbers can be reconstructed from boundary response operators.

The approach applies to both Maxwell and Dirac systems in anisotropic media.

Complete boundary knowledge allows full determination of topological invariants.

Abstract

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.