---
tag: 82ZPRD39J7
kind: concept
name: Chern–Weil homomorphism
msc: [53C05, 57R20]
sources:
  - url: https://en.wikipedia.org/wiki/Chern%E2%80%93Weil_homomorphism
same_as:
  wikidata: Q3087844
  wikipedia: Chern–Weil homomorphism
---

## Summary

The Chern–Weil homomorphism turns invariant polynomials on the Lie algebra of a group $G$ into de Rham cohomology classes.
These classes live on the base of a principal $G$-bundle.
One chooses a connection, plugs its [[curvature form|ZAJFW1S078]] into the polynomial, and obtains a closed form whose cohomology class does not depend on the connection.
The classes obtained are the real [[characteristic classes|7HACJC0JE2]] of the bundle, such as Chern and Pontryagin classes.
It links local differential geometry, through curvature, with global topology.

## Explanations

### L1 human

A bundle can be twisted in ways that no local measurement detects directly.
Curvature, however, is local and can be computed point by point.
Chern–Weil theory takes a symmetric expression in the curvature, such as a trace or a determinant, and integrates it or takes its cohomology class.
The answer is the same for every connection, so it measures the twisting of the bundle itself.
The Gauss–Bonnet theorem, where the integral of Gaussian curvature over a closed surface is $2\pi$ times the Euler characteristic, is the simplest instance.