Characterization of locally standard, Z-equivariantly formal manifolds in general position

We give a characterization of locally standard, Z-equivariantly formal manifolds in general position. In particular, we show that for dimension $2n$ at least $10$, to every such manifold with labeled GKM graph Γ there is an equivariantly formal torus manifold such that the restriction of the Tⁿ-action to a certain Tⁿ⁻¹-action yields the same labeled graph Γ, thus showing that the (equivariant) cohomology with Z-coefficients of those manifolds has the same description as that of equivariantly formal torus manifolds.