Non-neutrality of the Stiefel manifolds \(V_{n,k}\) (Q2367218)

The only nontrivial action \(\lambda\) of the orthogonal group \(\text{O}(k)\) on the Stiefel manifold \(V_{n,k}\), the set of \(k\) orthonormal column \(n\)-vectors, through right multiplication is induced by the self-map \(g\) changing the sign of any columns. \textit{I. M. James} calls \(V_{n,k}\) neutral if \(\lambda=1\), and shows that \(V_{2m,2s+1}\) is neutral and makes the conjecture: if \(V_{2m+1,2s}\) is neutral with \(m\geq 5\) then \(2m+2\) is a power of two. But he succeeded to verify this conjecture only for \(s=1,2\) or 4. The author in this paper proves that, if \(2m+2\) is not a power of two with \(m\geq s\), then \(V_{2m+1,2s}\) is non-neutral. In order to deduce his result, he constructs explicitly a homotopy \(1\simeq g: V_{2m,2k+1}\to V_{2m,2k+1}\) and examines a cohomological behavior of the \(Z_ 2\)-map \(\phi: V_{2m,2s+1}\to V_{2m-1,2s}X_{Z_ 2} S^ 1\) and finally use an argument leading to a contradiction.