Editing Pontryagin class (section)

== Definition ==
Given a real vector bundle <math>E</math> over <math>M</math>, its <math>k</math>-th Pontryagin class <math>p_k(E)</math> is defined as 
:<math>p_k(E) = p_k(E, \Z) = (-1)^k c_{2k}(E\otimes \Complex) \in  H^{4k}(M, \Z),</math>
where:
*<math>c_{2k}(E\otimes \Complex)</math> denotes the <math>2k</math>-th [[Chern class]] of the [[complexification]] <math>E\otimes \Complex = E\oplus iE</math> of <math>E</math>, 
*<math>H^{4k}(M, \Z)</math> is the <math>4k</math>-[[cohomology]] group of <math>M</math> with [[integer]] coefficients.

The rational Pontryagin class <math>p_k(E, \Q)</math> is defined to be the image of <math>p_k(E)</math> in <math>H^{4k}(M, \Q)</math>, the <math>4k</math>-cohomology group of <math>M</math> with [[Rational number|rational]] coefficients.