Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $Y$?

share|improve this question
add comment

1 Answer

up vote 4 down vote accepted

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.

share|improve this answer
2  
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. –  Peter May Nov 14 '13 at 2:55
    
@PeterMay of course, somehow that slipped my mind. I have corrected the attribution now. –  Vidit Nanda Nov 14 '13 at 4:03
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.