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Let $G$ be an abelian group (if this helps, let's say also finite). Let $BG_+$ be the classifying space together with a disjoint base point. What are the stable homotopy groups $$\pi^s_m (BG_+) := colim_n \pi_{n+m} (S^n \wedge BG_+)?$$ for example for $m=2$.

(The little I know is this: for $G=1$, we get the stable homotopy groups of spheres. For general $G$ and $m=0$, we get $\mathbf Z$, for $m=1$, we get $G/[G,G] \times \mathbf Z/2$.)

Thank you!

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    $\begingroup$ $BG_+$ splits stably into the wedge of its $p$-completions, for primes $p$ dividing the order of $G$. This gives a decomposition into a direct sum. However I think understanding the stable homotopy groups of $p$-completions of classifying spaces of finite groups is still an active area of research. See papers of Levi, Castellana, Crespo, Scherer, Viruel, and others. $\endgroup$ – Mark Grant Nov 22 '11 at 17:42
  • $\begingroup$ oh, I didn't know I'm touching deep waters here. However, is something known for say, for $G = \mathbf Z / 2$? $\endgroup$ – jakob Nov 23 '11 at 8:22
  • $\begingroup$ The Kahn-Priddy Theorem may be of interest to you. I think it says in this case that the $2$-torsion in the stable homotopy of $BG$ surjects onto the $2$-torsion in the stable homotopy of spheres. $\endgroup$ – Mark Grant Nov 24 '11 at 7:34

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