In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.

Given a real vector bundle ${\displaystyle E}$ over ${\displaystyle M}$, its ${\displaystyle k}$-th Pontryagin class ${\displaystyle p_{k}(E)}$ is defined as

where:

The rational Pontryagin class ${\displaystyle p_{k}(E,\mathbb {Q} )}$ is defined to be the image of ${\displaystyle p_{k}(E)}$ in ${\displaystyle H^{4k}(M,\mathbb {Q} )}$, the ${\displaystyle 4k}$-cohomology group of ${\displaystyle M}$ with rational coefficients.

The total Pontryagin class

is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e.,

for two vector bundles ${\displaystyle E}$ and ${\displaystyle F}$ over ${\displaystyle M}$. In terms of the individual Pontryagin classes ${\displaystyle p_{k}}$,

and so on.