$X_M$-Harmonic Cohomology and Equivariant Cohomology on Riemannian Manifolds With Boundary

Given a Riemannian manifold $M$ with boundary and a torus $G$ which acts by isometries on $M$ and let $X$ be in the Lie algebra of $G$ and corresponding vector field $X_M$ on $M$, we consider Witten's coboundary operator $\d_{X_M} = \d+\iota_{X_M}$ on invariant forms on $M$. In \cite{Our paper} we introduce the absolute $X_M$-cohomology $H^*_{X_M}(M)= H^*(\Omega^{*}_G,\,\d_{X_M})$ and the relative $X_M$-cohomology $H^*_{X_M}(M,\,\partial M)= H^*(\Omega^{*}_{G,D},\,\d_{X_M})$ where the $D$ is for Dirichlet boundary condition and $\Omega^{*}_G$ is the invariant forms on M. Let $\delta_{X_M}$ be the adjoint of $d_{X_M}$ and the resulting \emph{Witten-Hodge-Laplacian} is $\Delta_{X_M}= \d_{X_M}\delta_{X_M} + \delta_{X_M}\d_{X_M}$ where the space $\ker\Delta_{X_M}$ is called the $X_M$-harmonic forms. In this paper, we prove that the (even/odd) $X_M$-harmonic cohomology which is the $X_M$-cohomology of the subcomplex $(\ker\Delta_{X_M},\d_{X_M})$ of the complex $(\Omega^{*}_G,\d_{X_M})$ is enough to determine the total absolute and relative $X_M$-cohomology. As conclusion, we infer that the free part of the absolute and relative equivariant cohomology groups are determined by the (even/odd) $X_M$-harmonic cohomology when the set of zeros of the corresponding vector field $X_M$ is equal to the fixed point set $F$ for the $G$-action