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
http://dx.doi.org/10.1090/conm/058.2/893841
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