From: Greg Kuperberg
Subject: 2-sphere bundles of joy
Date: Mon, 20 Jul 1998 20:23:08 -0700 (PDT)
For some work in geometric topology (maybe only to convince myself that I
am not completely a librarian these days) I need to understand
real 4-plane and 3-plane bundles over (reasonable) topological spaces.
Specifically I think the following should be true:
An oriented orthogonal 4-plane bundle E over some space X is an
so(4)-bundle, and so(4) double-covers so(3) x so(3), so E yields two
oriented orthogonal 3-plane bundles Lambda^+ and Lambda^-. These two
auxiliary bundles can be defined as the self-dual and anti-self-dual
subbundles of Lambda^2(E), so their names make some sense. Now E has
characteristic classes
w_2 = w_2(E)
p_1 = p_1(E)
chi = chi(E)
Likewise Lambda^+ and Lambda^- have classes
p^+- = p_1(Lambda^+-)
w^+- = w_2(Lambda^+-).
I am relatively sure that the relation w^+ = w^- = w_2 holds, and
that it is the only relation required for a pair of 3-plane bundles
Lambda^+- to be realized by a 4-plane bundle E. I am less sure
of the following but I would also like it to be true:
p^+ = p_1 + 2*chi
p^- = p_1 - 2*chi
Finally oriented orthogonal 3-plane bundles. If you such a bundle F,
then there is an associated S^2 bundle. I would like to say that the
total space Y (i.e., the fibration is S^2 -> Y -> X) has a canonical
second cohomology class omega that restricts to [S^3] on each fiber.
omega should be integral if the bundle F is spin, but possibly only
half-integral if F is not spin. Moreover, I would like it to be
the case that if pi:Y -> X is the bundle map, that
omega^2 = pi^*(p_1(F))/4.
(Of course I understand that these statements about omega could just
as well be phrased as assertions about BSO(3)). Are all of these
things true?
Greg