Indecomposability of the Stiefel manifolds \(V_{m,k}\)
Let \(V_{m,k}\) denote the Stiefel manifold of orthogonal \(k\)-frames in \({\mathbb{R}}^ m\). Assume \(k>1\). Adams' solution of the vector fields problem for spheres is that the fibration \(V_{m-1,k-1}\rightarrow V_{m,k}\rightarrow S^{m-1}\) admits a cross section if and only if \(m\equiv0\mod 2^{\sigma(k)}\) where \(\sigma(k)\) is the number of \(0<s<k\) and \(s\equiv 0, 1, 2\) or \(4\pmod 8\). By definition this fibration is decomposable in the sense of James, if \(V_{m,k}\simeq V_{m-1,k- 1}\times S^{m-1}\). James has shown that such a decomposition implies \(m=2^ r\) for some \(r\geq\sigma(k)\) and furthermore if \(k\) is even, then \(m=2, 4, 8\). P. Selick has proved that \(V_{m,3}\) does not admit a nontrivial decomposition up to homotopy, in particular, \(V_{m,3}\not\simeq V_{m-1,2}\times S^{m-1}\) for \(m\neq 2, 4\hbox{ or }8\). The main result of the paper is: Theorem 1.1 Suppose \(m=2^{r+1}\). There does not exist a homotopy decomposition \(V_{m.k}\simeq V_{m-1,k-1}\times S^{m-1}\) for any \(k\geq3\) and \(r\) with \(r+1\geq\sigma(k)\) and \(r\geq 4\). The remaining unsettled problems are the cases \(V_{16,3}\), \(V_{16,5}\), \(V_{16,7}\). A space \(W\) is said to be homotopically decomposable if there are spaces \(X, Y\) with \(X\not\simeq *\), \(Y\not\simeq *\) and \(W\simeq X\times Y\). Homotopy decomposition at any prime \(p\) is similarly defined. The author obtains: Corollary 1.2 For \(m\neq 2, 4, 8 \hbox{ or }16\), the Stiefel manifold \(V_{m,k}\) for \(k<m\) is homotopically indecomposable at prime 2. These results are very deep. The proofs are hard and technical, using Adams secondary cohomology operations.
scientific article; zbMATH DE number 8861