= Surjectivity of special orthogonal stabilization below the sphere dimension
{title2=$\pi_r(SO(s))\twoheadrightarrow\pi_r(SO(s+1))\quad(r<s)$}
Project a based map $\alpha:S^r\to SO(s+1)$ to $S^s$. Since $r<s$, the resulting <homotopy class> vanishes. Lift a based null-homotopy by the <special orthogonal sphere fibration>. Its endpoint lies in the fiber $SO(s)$. If the lift moves its base point, right-multiply at time $t$ by the inverse of its base-point value, which lies in the fiber. This keeps the projected homotopy unchanged and makes the lift based. Thus $\alpha$ is based-homotopic to a map into $SO(s)$, proving the asserted surjectivity.
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