Contemporary Mathematics Volume 58, Part ll, 1987 THE K-THEORY OF PROJECTIVE STIEFEL MANIFOLDS Enrique Antoniano Here I want to state some results I recently obtained in collaboration with Samuel Gitler and Jack Ucci (theorem 6). about the K-theory of projective Stiefel manifolds. I will begin by explaining what these manifolds are and the reason of our interest in all this. If k n. the Stiefel manifold vn.k" consist of all orthonormal k-frames in Rn: Vn.k = {(vp···•vk)lvi e Rand Vi •Vj = 6ij} These manifolds are homogeneous spaces of the orthogonal group: V k ={left cosets of O(n-k)C O(n)} = O(n)/O(n-k) n. A ~ (~ ~) The projective Stiefel manifold X k is obtained by identifying each frame n. (v1•..••vk). with fts negative (-v1•.•••~vk) and it is also an homogeneous space of the orthogonal group: xn.k = O(n)/O(n-k) X z2 where z2"' {±I}CO(n) Now. we have a double covering vn.k + xn.k and this double covering is classified by a map: ... f: xn.k + p Consider the question suggested by the following diagram: ,X k S? , n .... -4-f pm, '"'-' p'"' Given n and k. what is the maximal value of m for which there exists a function s making the diagram commutative? This question is related to other known problems as we can see from the next two propositions see [1]. [4]: PROPOSITION 1: There is a map s making the preceeding diagram commutative if and only if there is a skew-linear and non singular map Rm+l x Rk + Rn 1 © 1987 American Mathematical Society 0271-4132/87 Sl.OO + 1.25 per page
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