Chern–Weil homomorphism

Builds on Connection on a principal bundle Curvature form de Rham cohomology Characteristic class

Definition

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ , and let $\mathbb{K}$ be $\R$ or $\C$ . For $k \ge 0$ , let $I^k(G)$ be the space of symmetric $k$ -linear maps $f : \mathfrak{g}^k \to \mathbb{K}$ that are invariant under the adjoint action: $f(\mathrm{Ad}_g X_1, \dots, \mathrm{Ad}_g X_k) = f(X_1, \dots, X_k)$ for all $g \in G$ . Let $\pi : P \to M$ be a principal $G$ -bundle with a connection form $\omega$ and curvature form $\Omega$ . For $f \in I^k(G)$ , the $2k$ -form $f(\Omega, \dots, \Omega)$ on $P$ (formed with the wedge product) equals $\pi^* \alpha_f$ for a unique closed $2k$ -form $\alpha_f$ on $M$ . The Chern–Weil homomorphism is the map $I^*(G) = \bigoplus_{k \ge 0} I^k(G) \to H^{\mathrm{even}}_{\mathrm{dR}}(M; \mathbb{K}), \quad f \mapsto [\alpha_f],$ into de Rham cohomology, which does not depend on the choice of $\omega$ .

Remarks

The form $f(\Omega, \dots, \Omega)$ descends to $M$ because $\Omega$ vanishes on vertical vectors and $f$ is $\mathrm{Ad}$ -invariant; it is closed by the Bianchi identity $d\Omega = [\Omega \wedge \omega]$ and invariance. For two connections $\omega_0, \omega_1$ , the forms $\alpha_f$ differ by the exterior derivative of a transgression form built from the path $\omega_t = (1-t)\omega_0 + t\omega_1$ ; so the class is independent of $\omega$ . With the product of symmetric multilinear forms, $I^*(G)$ is a commutative algebra and the map is an algebra homomorphism. Example: for $G = \mathrm{U}(1)$ , $\mathfrak{g} = i\R$ and $f(X) = \frac{i}{2\pi} X$ , the class $[\alpha_f]$ is the first Chern class of the associated line bundle. For $G = \mathrm{U}(n)$ , $\det\big(I + \frac{i}{2\pi} \Omega\big)$ gives the total Chern class, and its images are the characteristic classes with real coefficients.

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Characteristic class concept
Connection on a principal bundle concept
Curvature form concept
de Rham cohomology concept
Chern–Weil homomorphism definition

Tag	`FBFFS10WGS`
URI	`m2://FBFFS10WGS`
Kind	definition
Writing	W0 stub
MSC	53C05, 57R20
Defines	Chern–Weil homomorphism
Uses	Connection on a principal bundle, Curvature form, de Rham cohomology, Characteristic class
Changed	2 November 2026

Chern–Weil homomorphism

Definition

Remarks

Reading path

Further reading

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