most recent 30 from http://mathoverflow.net 2013-05-21T21:56:52Z http://mathoverflow.net/feeds/question/61043 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61043/the-stiefel-whitney-and-pontryagin-classes-of-so3-bundles Manuel 2011-04-08T10:24:23Z 2011-04-08T10:24:23Z

<p>Consider $SO(n)$ bundles over smooth manifolds. Then using the fact that the Stiefel-Whitney classes are the modulo 2 reductions of the Chern classes, one can prove $w_{2i}^2(E) = p_i(E) \bmod 2$. Now consider an $SO(3)$ bundle over a 4-manifold. Since the particular case I am studying concerns K3 surfaces, let us assume that $H^2(M,\mathbb{Z})$ contains no torsion. Then it is stated in various places that more is true:</p> <blockquote> <p>$w_2^2(E) = p_1(E) \bmod 4$.</p> </blockquote> <p>I may be missing something elementary here, but where does the mod 4 comes from?</p>