= Low homotopy groups of special orthogonal groups
{title2=$\pi_1SO(2)=\mathbb Z,\quad\pi_1SO(n)=\mathbb Z/2\ (n\geq3)$}
All the <special orthogonal groups> are connected, and their second <homotopy groups> vanish. The rotation circle gives $SO(2)\cong S^1$. Conjugation by <unit quaternions> gives the twofold <universal cover> $S^3\to SO(3)$, so $\pi_1SO(3)=\mathbb Z/2$ and $\pi_3SO(3)=\mathbb Z$. The <special orthogonal sphere fibration> $SO(n-1)\to SO(n)\to S^{n-1}$ propagates the first two <homotopy groups> for $n\geq4$. In dimension four its <exact sequence> has a zero map from the <torsion group> $\pi_4S^3=\mathbb Z/2$ to $\pi_3SO(3)=\mathbb Z$, giving $\pi_3SO(4)=\mathbb Z^2$. For $SO(1)$ all positive groups vanish, and $\pi_3SO(2)=0$.
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