Nonvanishing of the fifth special-unitary homotopy group

For $N=2$, the assertion is [F1]. For $N=3$, the LES of [F2] contains ${\pi }_5(SU(3))⟶{\pi }_5(S^5)\stackrel{\mathrm{\partial }}{\to }{\pi }_4(SU(2))\mathrm{.}$ By [F1], the target of $\mathrm{\partial }$ has two elements. The degree theorem [F4] gives ${\pi }_5(S^5)\cong \mathbb{Z}$, and every homomorphism from $\mathbb{Z}$ to a group of order two has kernel containing $2\mathbb{Z}$, which is nonzero. Exactness identifies that kernel with the image of ${\pi }_5(SU(3))$, so ${\pi }_5(SU(3))\mathrm{\ne }0$. This argument makes no surjectivity assumption on $\mathrm{\partial }$.

For $N\ge 4$, [F3] and the LES [F2] make the inclusion-induced map ${\pi }_5(SU(N-1))\to {\pi }_5(SU(N))$ an isomorphism: its preceding and following sphere groups are ${\pi }_6(S^{2N-1})$ and ${\pi }_5(S^{2N-1})$, both zero. Induction from the nonzero $N=3$ case proves nonvanishing for every $N\ge 4$, completing all cases.