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