Stable splittings of Stiefel manifolds
Let \(F={\mathbb{R}}\), \({\mathbb{C}}\) or \({\mathbb{H}}\) and let \(V_{n,q}\) denote the Stiefel manifold of q-frames in \(F^ q\). Identifying \(V_{n,q}\) with the space of Hermitian inner product preserving (right) F-linear maps from \(F^ q\) to \(F^ n\), the author defines a filtration by closed subsets \((\phi_ 0\) being fixed) \[ F_ k V_{n,q}=\{\phi | \dim_ F \ker (\phi +\phi_ 0)\geq q-k\}. \] Using the Bott-Morse functions of \textit{T. Frankel} [Differential and Combinat. Topology, Symp. Marston Morse, Princeton, 37-53 (1963; Zbl 0134.426)] the author identifies the strata \(F_ k-F_{k-1}\) as vector bundles and shows that there is a stable homotopy equivalence \(F_ k\simeq \bigvee_{j\leq k}F_ j/F_{j-1}\) in which, of course, the wedge factors are Thom spaces. A particular case of the result is a stable splitting for the classical groups \(O_ n\), \(U_ n\), \(Sp_ n\), \(SO_ n\) or \(SU_ n\). Such a splitting was originally conjectured by C. A. McGibbon.
stable splitting for the classical groups
scientific article; zbMATH DE number 3929943