Unordered Stiefel bundle

For a rank-$n$ Euclidean vector bundle, take its ordered orthonormal $k$-frame bundle and quotient by the symmetric group permuting the $k$ vectors. The whole fiber is generally positive-dimensional. A prescribed global framing selects a finite subcover consisting of the unordered $k$-element subsets of its distinguished frame vectors.

A global ordered framing selects the displayed $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$-sheeted cover inside the unordered Stiefel bundle. It is a trivial cover because the frame labels are global. Selecting subsets of a direct-sum framing decomposes the cover as the disjoint union of fiber products of the two summands' subset covers.