Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology

In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Lambda on a compact Riemannian manifold M with boundary partial derivative M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field X-M on M. Witten defines an inhomogeneous coboundary operator d(XM) = d + l(XM), on invariant forms on M. The main purpose is to adapt Belishev-Sharafutdinov's boundary data to invariant forms in terms of the operator d(XM) in order to investigate to what extent the equivariant topology of a manifold is determined by the corresponding variant of the DN map. We define an operator Lambda(XM) on invariant forms on the boundary which we call the X-M-DN map and using this we recover the X-M-cohomology groups from the generalized boundary data (partial derivative M. Lambda(XM)). This shows that for a Zariski-open subset of the Lie algebra, Lambda(XM) determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of X-M-cohomology groups from Lambda(XM). These results explain to what extent the equivariant topology of the manifold in question is determined by Lambda(XM). (C) 2011 Elsevier B.V. All rights reserved.