---
tag: BJX6QS0ANT
kind: concept
name: Euler class
msc: [57R20, 55R40, 55R25]
specializes: [7HACJC0JE2]
sources:
  - url: https://en.wikipedia.org/wiki/Euler_class
same_as:
  wikidata: Q2193452
  wikipedia: Euler class
---

## Summary

The Euler class $e(E) \in H^n(B; \Z)$ is a [[characteristic class|7HACJC0JE2]] of oriented real vector bundles $E \to B$ of rank $n$.
It is the first obstruction to a nowhere-zero section: if $E$ has a section that never vanishes, then $e(E) = 0$.
For the tangent bundle of a closed oriented manifold, its value on the [[fundamental class|ZEA0PEM3B2]] is the [[Euler characteristic|RNDDWDAFJ5]].
It appears in the [[Gysin sequence|QJSPWM9F9P]] and is defined from the Thom class of the [[Thom isomorphism|N685RXHDZX]].

## Explanations

### L1 human

The Euler class counts, with signs, the zeros that a section of a bundle is forced to have.
For the tangent bundle of a surface, a section is a field of arrows, and the count is the Euler characteristic of the surface.
On the sphere the count is $2$, so every field of arrows on the sphere has a zero somewhere; this is the hairy ball theorem.
On the torus the count is $0$, and indeed the torus carries a field of arrows with no zeros.