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In ordinary algebraic topology the Adams spectral sequence can be applied for any cohomology theory $E$ and in good cases it converges to the stable homotopy classes of maps (of the E-nilpotent completion) of some chosen spaces $X,Y$. I guess that similar results are know for equivariant cohomology theories. I am looking for something like a generalization of Greenlees "Stable Maps into Free $G$-Spaces".

Q.: Is it true that such more general statements exist? Where can I find them?

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These kinds of things also exist in equivariant homotopy theory. However, there are some fairly significant issues. One is that, because the homotopy groups take values in Mackey functors rather than just abelian groups, it is harder to understand what kind of completion one expects based on E. A second is that homological algebra of Mackey functors is very hard. A third is more serious, and that is that it is much, much harder to compute the analogue of the Steenrod algebra $E^* E$, it is less likely to be "flat" over E, and so the spectral sequence is difficult to compute in practice. – Tyler Lawson Mar 22 '10 at 12:57
However, in the positive direction there's been a recent burst of work in computations in motivic stable homotopy theory, which very much has the flavor of Z/2-equivariant stable homotopy theory. – Tyler Lawson Mar 22 '10 at 13:04
i recall hearing references to work of araki and iriye, but it may be a bad source... Like tyler says, people often use the equivariant results as a guide to see what to do motivicly, or at least i know one person that has done this, so certainly there is some connection. i think this might come from having two different models for spheres, one with trivial action one with the antipodal action. this is a guess. i will look for the araki-reference. – Sean Tilson Apr 2 '10 at 5:27
I know this Araki/Iriye paper on stable homotopy groups of spheres with involution a little bit, but I am pretty sure they didn't use ASS nor can I remember any spectral sequence in their paper. Thanks Tyler and Sean. Any references in this direction? – user2146 Apr 2 '10 at 8:57

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