  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 $\Lambda$ 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 $\Lambda$. This shows that the DN map $\Lambda$
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 $d_{X_M} = d+\iota_{X_M}$ 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 $\delta_{X_M}$. In other words, we define an operator
$\Lambda_{X_M}$ 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 $\Lambda_{X_M}$
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 $\Lambda_{X_M}$. These results explain to what
extent the equivariant topology of the manifold in question is determined by
the $X_M$-DN map $\Lambda_{X_M}$. Finally, we illustrate the connection between
Belishev and Sharafutdinov's boundary data on $\partial F$ and ours on
$\partial M$.
