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Stiefel manifold
\(V_n(X)\) is the Stiefel manifold of dimension \(n\) orthonormal frames in \(X\).
Stiefel manifold
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examples of classifying spaces
\(\operatorname{GL}_n({\mathbf{R}}) \to V_n \to {\mathbf{B}}\operatorname{GL}_n({\mathbf{R}}) \cong {\operatorname{Gr}}_n({\mathbf{R}}^\infty)\) for \(V_n\) the Stiefel manifold.