Timeline for Why are equivariant homotopy groups not RO(G)-graded?

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May 17, 2011 at 0:17	comment	added	Tom Goodwillie		Note that the basic example $G_+$ (i.e. $(G/H\times D^n)/(G/H\times S^{n-1})$ in the case $H=1$, $n=0$) admits no nontrivial maps from any representation sphere if $G$ is nontrivial. In the stable setting it's not quite that easy to see that representation spheres are inadequate for detecting weak equivalences. They are adequate when $G$ has order $2$ because the homotopy cofiber of $G_+\to S^0$ is a representation sphere. But things go wrong already when $G$ has order $3$.
May 17, 2011 at 0:11	comment	added	Tom Goodwillie		They are a natural thing to turn to because when you start looking at some obvious classes of $G$-spaces such as those admitting an ordinary CW structure in which the $G$-action permutes the cells, or smooth manifolds with a smooth $G$-action, they tend to be things you can build up in the way Charles describes.
May 16, 2011 at 22:44	comment	added	Dylan Wilson		Are these cells somehow more natural than some analog with representation spheres? Although I guess it's not obvious what the analog of the disk would be?
May 16, 2011 at 22:35	history	answered	Charles Rezk	CC BY-SA 3.0