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Abstract: On a compact Riemannian manifold with boundary, the absolute and relativecohomology groups appear as certain subspaces of harmonic forms. DeTurck andGluck showed that these concrete realizations of the cohomology groupsdecompose into orthogonal subspaces corresponding to cohomology coming from theinterior and boundary of the manifold. The principal angles between theseinterior subspaces are all acute and are called Poincare duality angles. Thispaper determines the Poincare duality angles of a collection of interestingmanifolds with boundary derived from complex projective spaces and fromGrassmannians, providing evidence that the Poincare duality angles measure, insome sense, how -close- a manifold is to being closed.This paper also elucidates a connection between the Poincare duality anglesand the Dirichlet-to-Neumann operator for differential forms, which generalizesthe classical Dirichlet-to-Neumann map arising in the problem of ElectricalImpedance Tomography. Specifically, the Poincare duality angles are essentiallythe eigenvalues of a related operator, the Hilbert transform for differentialforms. This connection is then exploited to partially resolve a question ofBelishev and Sharafutdinov about whether the Dirichlet-to-Neumann mapdetermines the cup product structure on a manifold with boundary.



Author: Clayton Shonkwiler

Source: https://arxiv.org/







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