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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

MSC:
 55P40 Suspensions 55P42 Stable homotopy theory, spectra
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