---
tag: 7HACJC0JE2
kind: concept
name: Characteristic class
msc: [55R40, 57R20]
sources:
  - url: https://en.wikipedia.org/wiki/Characteristic_class
same_as:
  wikidata: Q1062978
  wikipedia: Characteristic class
---

## Summary

A characteristic class assigns to each bundle of a given type over a space $X$ a cohomology class of $X$, compatibly with pullbacks.
It measures how far a bundle is from being trivial: the trivial bundle has only the trivial classes.
The main families are the [[Stiefel–Whitney classes|JQCK7A210W]], the Chern classes, the [[Pontryagin classes|DN11FKWN63]] and the [[Euler class|BJX6QS0ANT]].
Through the [[classifying space|WSZ40H5E4T]], characteristic classes of principal $G$-bundles correspond to cohomology classes of $BG$.

## Explanations

### L1 human

Two bundles over the same space can look alike locally but be twisted in different ways globally.
A characteristic class is a label in the cohomology of the base that detects such twisting.
The labels behave well under change of base: pulling a bundle back along a map pulls the label back too.
For example, the first Stiefel–Whitney class of the Möbius band over the circle is nonzero, while that of the cylinder is zero, so the two bundles differ.