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 $Y$?
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asked Nov 14, 2013 at 1:41
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1 Answer
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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 homotopy-inverse.
You can find all this and much more in John Greenlees and Peter May's lovely notes here.
answered Nov 14, 2013 at 1:55
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2$\begingroup$ Thank you very much for the nice compliment, but I would like to emphasize that the notes you refer to are joint with John Greenlees. $\endgroup$ Commented Nov 14, 2013 at 2:55
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$\begingroup$ @PeterMay of course, somehow that slipped my mind. I have corrected the attribution now. $\endgroup$ Commented Nov 14, 2013 at 4:03