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What are the stable homotopy groups of the symplectic groups $Sp(2n)$? Is there a reference which contains a detailed treatment?

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The stable range for $\pi_k(\mathrm{Sp}(2n))$ is $k \leq 8n + 1$. In this case we have $$\pi_k(\mathrm{Sp}(2n)) \cong \begin{cases} \Bbb Z, & k \equiv 3, 7 \pmod 8, \\ \Bbb Z/2, & k = 4, 5 \pmod 8, \\ 0, & k \equiv 0, 1, 2, 6 \pmod 8. \end{cases}$$ This was first determined by Bott:

Bott, Raoul. The stable homotopy of the classical groups. Annals of Mathematics, Second Series, Vol. 70, No. 2 (Sep., 1959), pp. 313-337.

For more about homotopy groups of $\mathrm{Sp}(2n)$, see the following article:

Mimura, Mamoru and Toda, Hiroshi. Homotopy groups of symplectic groups. J. Math. Kyoto Univ. Volume 3, Number 2 (1963), 251-273.

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