Subject: Question about classification of vector/projective bundles over
surfaces
From: André Gama Oliveira
Date: Sun, 22 Jan 2006 12:35:50 -0000
My name is Andre, I'm from Portugal, and I have two question to post,
if possible, to the Algebraic Topology discussion list.
I would like to have a reference for a proof of the following fact:
The (first and second) Stiefel-Whitney classes completely classify
continuous real vector bundles of rank n>=3 over (real) surfaces
of genus higher than zero.
(In fact, the same is true for any 2-dimensional CW-complex.)
Do you know where can I find it? I have already tried to find
a proof, but I wasn't able to.
I know that if two vector bundles, over such a surface, have
the first Stiefel-Whitney class (w1) equal to zero and with
the same second Stiefel-Whitney class (w2), then they are
homeomorphic (in this case there is a reduction to SO(n) which
is connected and so the classification is given by the fundamental
group of SO(n), or equivalently, by w2) . But what if they have the
same w1 and w2, with w1 not zero. Can I still conclude that they are
homeomorphic? They are homeomorphic over the 1-skeleton of the surface.
But is there a homeomorphism over all the surface?
On the other hand, I would also like to know a reference for the
classification of real projective bundles of rank >2 over real
surfaces of genus >0. This case seems to be more complicated because
BPO(n) is not 2-simple (i.e., pi_1(BPO(n)) does not act trivially on
pi_2(BPO(n))), so I might have to use local coefficients to have the
invariants. There is one invariant which is the obstruction to the
reduction to PSO(n). Then there are other invariants which are the
obstructions to the lifting to O(n) and then to Pin(n). These must
be in the orbit space of action of pi_0(PO(n)) on pi_1(PO(n)), which
has three elements. Note that pi_1(PO(n)) is not central in Pin(n),
the universal cover of PO(n). Does someone knows how to do it?
Thank you in advance,
Andre.