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James numbers. (English) Zbl 0627.57026

Let \(W_{n,k}\) denote the complex Stiefel manifold of orthogonal k- frames in \({\mathbb{C}}^ n\) and \(q: W_{n,k}\to W_{n,1}=S^{2n-1}\) the canonical projection. Define the unstable James number Ū(n,k) to be the index of \(q_ *(\pi _{2n-1}(W_{n,k}))\) in \(\pi _{2n-1}(W_{n,1})\) and the stable James number U(n,k) similarly by replacing homotopy by stable homotopy. Then U(n,k) divides Ū(n,k) and in the stable range \(n\geq 2k-1\) the two numbers are equal. The complete computation of these numbers is still an unsolved problem.
In a series of papers the authors obtained results closely related to James numbers but set up in a different context. These results allow the computation of U(n,k) in a large range of dimensions. The present paper, which may be read as an introduction to this series, contains the basic material on James numbers, a description and translation of the results of the other papers in the series as they apply to James numbers, and some applications of these computations exemplifying the geometrical significance of James numbers.

MSC:

57T20 Homotopy groups of topological groups and homogeneous spaces
55Q52 Homotopy groups of special spaces
55Q10 Stable homotopy groups
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References:

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