Let $T$ be a real torus, and let $X$ and $Y$ be $T$spaces. Under what conditions (if any) will the existence of graded $H^*_T$algebra isomorphism between the $T$equivariant cohomologies of $X$ and $Y$ (say over the rationals) imply the existence of a $T$equivariant homotopy equivalence between $X$ and $Y$?
One set of sufficient conditions may be obtained if your map plays nicely with respect to subspaces fixed by closed subgroups. Let $X$ and $Y$ be $G$spaces for any $G$ (not only the torus) and assume that you have an equivariant map $f:X \to Y$. If for any closed subgroup $H < G$ the induced map $X^H \to Y^H$ of $H$stable subspaces induces homotopy equivalence, then in fact $f$ admits an equivariant homotopyinverse. You can find all this and much more in John Greenlees and Peter May's lovely notes here. 

