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

In a recent paper, Belishev and Sharafutdinov consider a compact Riemannian manifold $M$ with boundary $\partial M$. They define a generalized Dirichlet to Neumann (DN) operator $Λ$ on all forms on the boundary and they prove that the real additive de Rham cohomology structure of the manifold in question is completely determined by $Λ$. This shows that the DN map $Λ$ inscribes into the list of objects of algebraic topology. 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 $M$, one defines Witten's inhomogeneous coboundary operator $d_{X_M} = d+ι_{X_M}$ on invariant forms on $M$. The main purpose is to adapt Belishev and Sharafutdinov's boundary data to invariant forms in terms of the operator $d_{X_M}$ and its adjoint $δ_{X_M}$. In other words, 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 long exact $X_M$-cohomology sequence of the topological pair $(M,\partial M)$ from an isomorphism with the long exact sequence formed from our boundary data. We then show that $Λ_{X_M}$ completely determines the free part of the relative and absolute equivariant cohomology groups of $M$ when the set of zeros of the corresponding vector field $X_M$ is equal to the fixed point set $F$ for the $G$-action. In addition, we partially determine the mixed cup product (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 the $X_M$-DN map $Λ_{X_M}$. Finally, we illustrate the connection between Belishev and Sharafutdinov's boundary data on $\partial F$ and ours on $\partial M$.

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