Subject: Re: question about vector bundles
From: Johannes Ebert
Date: Mon, 23 Jan 2006 16:31:59 +0100 (CET)
Hello Andre,
I can answer your first question. You want to know
that if E_1, E_2 are real vector bundles on a
2-dimensional CW complex X with the same w_1 and w_2
and the same dimension n, then they are isomorphic.
I do not know a reference for this, but a proof is
not difficult.
Let (w_1,w_2): BO(n) \to K(Z/2;1) \times K(Z/2;2)
be the universal Stiefel-Whitney class. The
Stiefel-Whitney classes of the two vector bundles
in question agree and they give a map
\phi: X \to K(Z/2,1) \times K(Z/2;2).
The classifying maps f_1, f_2:X \to BO(n)
for E_1 and E_2 are two liftings of
\phi:(w_1,w_2) \circ f_i = \phi.
You want to know that both lifts are homotopic.
This can be seen by
obstruction theory, because the homotopy fiber F
of (w_1, w_2) is 2-connected (it is even 3-connected).
The fact that BO(n) is not a simple space does not matter here.
That F is 3-connected, follows because (w_1,w_2) induces
isomorphisms \pi_i(BO(n)) \to \pi_i(K(Z/2;1)\times K(Z/2;2))
for i=0,1,2,3.
Best regards,
Johannes Ebert