Subject: Re: question about vector bundles
From: Johannes Ebert <ebert@math.uni-bonn.de>
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