Natural numbers $n$ for which $\pi_n(G)=0$ for all Lie groups

Question

It is a classical theorem that $\pi_2(G)$ is trivial for all Lie group $G$.

Is $n=2$ the only natural number with the property that $\pi_n(G)$ is trivial for every Lie group $G$?

Answer 1

Yes. For all $n>2$, $\pi_n(S^3)$ is nontrivial: see https://arxiv.org/abs/1506.00952.

Or, using just results from the 1950s, by Bott periodicity, $\pi_n(U(m))\cong\mathbb{Z}$ if $n$ is odd and $m$ is sufficiently large. Even $n>2$ are handled by the fact that $\pi_{2m}(U(m))\cong\mathbb{Z}/(m!)$ for all $m$.