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 Λ on a compact Riemannian manifold M with boundary ∂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 X M = d + ι X M on invariant forms on M. The main purpose is to adapt Belishev–Sharafutdinovʼs boundary data to invariant forms in terms of the operator d X M 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 Λ X M 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 ( ∂ M , Λ X M ) . This shows that for a Zariski-open subset of the Lie algebra, Λ X M 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 Λ X M . These results explain to what extent the equivariant topology of the manifold in question is determined by Λ X M .