Non-neutrality of the Stiefel manifolds \(V_{n,k}. \) II (Q5956080)

For Part I see [the author, ibid. 32, No. 1, 105-120 (1993; Zbl 0776.55009)]. NEWLINENEWLINENEWLINEEach point of the Stiefel manifold \(V(n,k)\) of orthogonal \(k\)-frames in Euclidean \(n\)-space can be regarded as an \(n\times k\) matrix. The orthogonal group \(O(k)\) acts on \(V(n,k)\) by matrix multiplication and so each element of \(O(k)\) gives rise to a self-map of \(V(n,k)\). If this element is in \(SO(k)\), then the homotopy class of the self-map is that of the identity. The elements of \(O(k)- SO(k)\) also give self-maps which all lie in one homotopy class. The Stiefel manifold \(V(n,k)\) is said to be neutral if this homotopy class is also that of the identity. The neutrality problem on \(V(n,k)\) is to determine for what \(n\) and \(k\), \(V(n,k)\) is neutral. The author proves the following result. NEWLINENEWLINENEWLINETheorem. For \(m\) greater than or equal to 5, \(V(2n-1,k)\) is non-neutral where \(k=2t\) is greater than or equal to \(n + 2\) and less than \(2n-2\). The author conjectures that the result is also true for \(m=4\). The problem for \(k = 2\) is connected to a problem in the homotopy theory of spheres. The details of the proof of this theorem are very involved.