Poincaré duality angles and the Dirichlet-to-Neumann operator (Q2852309)

The main purpose of this paper is to show that the relative positions of cohomology groups can not only be recovered from boundary data, but in fact from boundary data which naturally arises in a significant inverse problem. Specifically, these relative positions are determined by the Cauchy data of harmonic differential forms, which can be represented as the graph of the Dirichlet-to-Neumann operator for differential forms. The paper also gives a partial answer to a question of Belishev and Sharafutdinov about whether the Cauchy data for differential forms determines the cup product structure on a manifold with boundary.