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Suspensions of Stiefel manifolds. (English) Zbl 0555.55010
Let \(V_{n,k}\) denote the Stiefel manifold of orthogonal k-frames in n- space and let \(CV_{n,k}\) denote its complex analogue. Let \(P_{n,k}={\mathbb{R}}P^{n-1}/{\mathbb{R}}P^{n-k-1}\) and let \(CP_{n,k}={\mathbb{C}}P^{n-1}/{\mathbb{C}}P^{n-k-1}.\) There are inclusions \(P_{n,k}\to V_{n,k}\) and \(\Sigma CP_{n,k}\to CV_{n,k}\) which are stable retracts. Let r(n,k) denote the least r such that \(\Sigma^ rP_{n,k}\) is a retract of \(\Sigma^ rV_{n,k}\) and similarly r(n,k,\({\mathbb{C}})\) in the complex case. The authors obtain bounds on these numbers, thereby answering some questions posed by I. M. James [The topology of Stiefel manifolds, Lond. Math. Soc. Lect. Note Ser. 24 (1976; Zbl 0337.55017)].
Reviewer: V.Snaith

55P40 Suspensions
55P42 Stable homotopy theory, spectra
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