Subject: Re: question about characteristic classes From: nkitchlo@math.ucsd.edu Date: Fri, 22 Dec 2006 02:33:31 -0800 (PST) Dear Wolfgang, Information regarding w_4 is retained in the Steenrod structure of the mod-2 cohomology ring (of the total space E). So for example, if we take the (unique) class in degree 3 and apply the second Steenrod square Sq2, then the answer is zero if and only if w_4 for the bundle is zero. This can indeed be shown using transgressions in the spectral sequence for the fibration E -> CP2 -> BSO(k). Hope this helps. Nitu. >> Subject: top question >> From: Wolfgang Ziller >> Date: Mon, 18 Dec 2006 23:10:02 -0500 >> >> Dear topology friends, >> >> I have an algebraic topology question which has applications in geometry >> (nonnegative curvature) >> It is a very concrete and special situation I am in. >> >> I have a family of SO(k) principal bundles P over CP2 , k>4 , >> with w_2 \ne 0 >> P arises from a concrete construction. >> All I know about P though is that SO(k) acts freely and I know the >> cohomology >> ring of P. From this knowledge can I can determine the bundle? >> By Dold Whitney's classification of bundles over simply connected >> 4-manifolds, >> such bundles are classified by w_2 , w_4 and p_1. >> p_1 can be interpreted as an integer by evaluating on a fundamental cycle. >> >> The structure group of course reduces to 4 dim bundles where w_4 is >> euler class >> mod 2. >> But my bundles have k>4 and from their construction there is no easy way >> to >> reduce the structure group explicitly. >> >From this though one can determine the restriction that need to be >> satisfied for >> these numbers since >> p_1=2k+2l e=k-l if w_2 = 0 and p_1 = 2k+2l+1 , e= k-l if w_2 \ne 0 >> where k,l >> are arbitrary integers. >> >> It is not hard to recognize w_2 and | p_1 | in the top of P : >> w_2\ne 0 iff P is simply connected and if p_1 \ne 0 then H4(P,Z) is a >> finite >> group of order |p_1|. >> >> If w_2=0 I am ok since then p_1= 2 mod 4 implies w_4 \ne 0 and p_1 = 0 mod >> 4 >> implies w_4 = 0 >> >> But when I have w_2 \ne 0 it only implies that >> if p_1 = 1 mod 4 then w_4 \ne 0 and if p_1 = 3 mod 4 then w_4=0. >> >> So my problem is that I cannot recognize the sign of p_1 from the coh ring >> . >> Is there some other information in the coh ring that will tell me what >> w_4 is? >> I assume that this is somehow encoded in the spectral sequence for the >> universal bundle (maybe with Z_2 coefficients). >> This is how one sees the above facts about w_2 and |p_1| for example >> >> Another question would be if you know two such bundles where p_1 >> differs by a >> sign concretely enough so I can see what to expect from their cohomology >> ring? >> >> Any advice would be appreciated.