The Stiefel-Whitney and Pontryagin classes of SO(3)-bundles - MathOverflow 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 The Stiefel-Whitney and Pontryagin classes of SO(3)-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>