Characteristic class

Let $\pi:P\to M$ be a principal G-bundle over a smooth manifold $M$. Fix an Ad-invariant polynomial $P$ on the Lie algebra $\mathfrak{g}$ of $G$.

Choose any principal connection on $P$, form its Chern–Weil form $\mathrm{CW}_P(\omega)\in \Omega^{2k}(M)$ as in the Chern–Weil form construction , and take its de Rham class:

This cohomology class is called the characteristic class associated to the invariant polynomial $P$ (and the bundle $P$). It is well-defined because:

• $\mathrm{CW}_P(\omega)$ is closed, and
• the cohomology class $[\,\mathrm{CW}_P(\omega)\,]$ is independent of the choice of connection $\omega$.

Characteristic classes are natural under pullback of bundles: if $f:N\to M$ is smooth and $f^*P\to N$ is the pulled-back bundle, then the characteristic class of $f^*P$ is $f^*c_P(P)$.

Examples

1. First Chern class of a circle bundle. For a principal $U(1)$-bundle, the de Rham class of the curvature form (with conventional normalization) gives the first Chern class $c_1\in H^2_{\mathrm{dR}}(M)$.

2. Pontryagin classes. For an $SO(n)$-bundle, Ad-invariant polynomials built from traces of even powers (e.g. $\mathrm{tr}(X^2)$, $\mathrm{tr}(X^4)$, etc.) produce the Pontryagin classes $p_i\in H^{4i}_{\mathrm{dR}}(M)$ via Chern–Weil theory.

3. Euler class. For an oriented $SO(2m)$-bundle, the Pfaffian polynomial yields a top-degree class in $H^{2m}_{\mathrm{dR}}(M)$, the Euler class; integrating a representative over a closed base manifold recovers the Euler number.