Special orthogonal sphere fibration
= Special orthogonal sphere fibration
{title2=$SO(s)\to SO(s+1)\to S^s$}
The last-column map $p(A)=Ae_{s+1}$ has fiber $SO(s)$, included as $\operatorname{diag}(B,1)$. Local completion of a unit vector by the <Gram-Schmidt process> gives local <fiber bundle> charts. Consequently it has the <homotopy lifting property> and the <long exact sequence of homotopy groups of a fibration>.