Pontryagin class ($p_k(E)=(-1{)}^k{c}_{2k}(E{\otimes }_{\mathbb{R}}\mathbb{C})$)

For a real vector bundle $E$, its Pontryagin classes areThey use the even-index Chern classes of the complexification. An integral qualification matters for direct sums: for the total Pontryagin class,Thus the usual multiplicative formula holds over the rational numbers, over the real numbers, or after discarding elements of order two. It need not hold without qualification in integral cohomology. The missing terms are products of odd Chern classes of complexified real vector bundles, which are annihilated by two.^[ref]

For a metric connection on a vector bundle with curvature matrix $R$, a real representative isThe Bianchi identity makes this a closed differential form, and Chern-Weil connection transgression makes its de Rham cohomology class independent of the connection on a vector bundle. This normalization gives the degree-four A-hat form $-p_1/24$. Curvature representatives record real classes and therefore cannot detect the integral torsion qualification.