---
tag: DN11FKWN63
kind: concept
name: Pontryagin class
msc: [57R20, 55R40]
specializes: [7HACJC0JE2]
sources:
  - url: https://en.wikipedia.org/wiki/Pontryagin_class
same_as:
  wikidata: Q40007
  wikipedia: Pontryagin class
---

## Summary

The Pontryagin classes $p_i(E) \in H^{4i}(B; \Z)$ are [[characteristic classes|7HACJC0JE2]] of real vector bundles $E \to B$ with integer coefficients.
They are defined from the Chern classes of the complexification $E \otimes \C$.
Pontryagin classes of the tangent bundle of a manifold give the Pontryagin numbers, which are invariants of oriented cobordism.
They enter the Hirzebruch signature theorem and Milnor's construction of exotic spheres.

## Explanations

### L1 human

A real vector bundle can be turned into a complex one by allowing complex scalars in each fiber.
Complex bundles have Chern classes, which are integer cohomology classes.
The Pontryagin classes are the Chern classes of this complexified bundle in degrees divisible by four, with a sign.
For the complex projective plane, the first Pontryagin class is three times the square of the generator, and this three is the reason its signature is one.