Homotopy groups for homogeneous spaces $SU(2n)/Sp(n)$ and $SU(2n)/SO(2n)$

Question

I would need to know $\pi_4$ and $\pi_5$ for $SU(2n)/Sp(n)$ and $SU(2n)/SO(2n)$ with $n\geq2$. The case of $SU(4)/Sp(2)$ is most important to me. I tried to search the literature but haven't been able to find the answer. Any hint or link to a good source would be highly appreciated!

Answer 1

Note that $SU(4) = Spin(6)$ and $Sp(2) = Spin(5)$ where $Spin(m)$ is the double cover of $SO(m)$. So

$$\frac{SU(4)}{Sp(2)} = \frac{Spin(6)}{Spin(5)} = \frac{SO(6)}{SO(5)} = S^5.$$

So $\pi_4(SU(4)/Sp(2)) = 0$ and $\pi_5(SU(4)/Sp(2)) = \mathbb{Z}$.