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.