# Stable homotopy groups of $Sp(2n)$

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

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