Is this group known outside of the stable range? If so, what is it? If not, what is known about it?
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$\begingroup$ And of course, you are concerned with the case of k even as for k odd `the answer' is obvious! You are really asking what is known about the complex Steifel manifold, which in stable range looks like the stuned projective space $\mathbb{C} P_k$. I suppose some papers of Mahowald will have answer to this. I presume, a complete answer to this, will solve many problems in immersion theory. $\endgroup$– user51223Commented May 28, 2015 at 20:55
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For the homotopy groups of $U(r)$, hence also of $BU(r)$, you can look at the survey by M. Mimura, "Homotopy theory of Lie groups", which is chapter 19 in the "Handbook of Algebraic Topology", edited by I. M. James. It gives $\pi_k(SU(n))$ for $k\le15$ and for $k\le 2n+6$, in section 3. The survey also lists many further references.
