Indecomposability of the Stiefel manifolds \(V_{m,3}\)
Denote by \(V_{m,k}\) the Stiefel manifold of orthonormal k-frames in \({\mathbb{R}}^ m\). Extending results of I. M. James the author proves that there is no homotopy equivalence between \(V_{m,3}\) and \(V_{m- 1,2}\times S^{m-1}\) for \(m=4,8,16\). For \(m=4,8\) such a homotopy equivalence exists. ``The case of \(V_{16,3}\) is left unsettled.'' ``The proof consists of showing that such a product decomposition... implies the existence of an element \(\vartheta\) of Arf invariant 1 in \(\pi^ S_{m-2}\) with the property that \(\eta\) \(\vartheta\) is divisible by 2, where \(\eta\) is the nontrivial element in the stable 1-stem.'' By results of H. E. A. Campbell, F. R. Cohen, F. P. Peterson and the author the statement follows.
scientific article; zbMATH DE number 4128074