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On complex Stiefel manifolds

Published online by Cambridge University Press:  24 October 2008

M. F. Atiyah  and
J. A. Todd
M. F. Atiyah
Affiliation:
Pembroke CollegeCambridge
J. A. Todd
Affiliation:
Downing CollegeCambridge
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Extract

In a recent series of papers (10), (11), (12), I. M. James has made an illuminating study of Stiefel manifolds. We shall begin by describing his results (for the complex case). Let Wn, k, for k > 1, denote the complex Stiefel manifold U(n)/U(n − k), where U(n) is the unitary group in n variables. Then we have a natural fibre map Wn, k → Wn, 1 = S2n−1, where Sr denotes the r-dimensional sphere. Let Pn, k, for k ≥ 1, denote the ‘stunted complex projective space’ obtained from the (n − 1)-dimensional complex projective space† Pn by identifying to a point a subspace Pn−k. Then we have a natural ‘cofibre map’ Pn, k → Pn, 1 = S2n−2. The space Pn, k is said to be S-reducible if some suspension of the map Pn, k → S2n−2 has a right homotopy inverse. The results of James can then be summarized as follows.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 56 , Issue 4 , October 1960 , pp. 342 - 353
Copyright
Copyright © Cambridge Philosophical Society 1960

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References

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