# Pontryagin class

In mathematics, the **Pontryagin classes**, named for Lev Pontryagin, are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four. It applies to real vector bundles.

## Definition

Given a real vector bundle *E* over *M*, its *k*-th Pontryagin class *p _{k}*(

*E*) is defined as

*p*(_{k}*E*) =*p*(_{k}*E*,**Z**) = (−1)^{k}*c*_{2k}(*E*⊗**C**) ∈*H*^{4k}(*M*,**Z**),

where:

*c*_{2k}(*E*⊗**C**) denotes the 2*k*-th Chern class of the complexification*E*⊗**C**=*E*⊕*iE*of*E*,*H*^{4k}(*M*,**Z**) is the 4*k*-cohomology group of*M*with integer coefficients.

The rational Pontryagin class *p _{k}*(

*E*,

**Q**) is defined to be the image of

*p*(

_{k}*E*) in

*H*

^{4k}(

*M*,

**Q**), the 4

*k*-cohomology group of

*M*with rational coefficients.

## Properties

The **total Pontryagin class**

is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e.,

for two vector bundles *E* and *F* over *M*. In terms of the individual Pontryagin classes *p _{k}*,

and so on.

The vanishing of the Pontryagin classes and Stiefel-Whitney classes of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to vector bundle isomorphism, there is a unique nontrivial rank 10 vector bundle *E*_{10} over the 9-sphere. (The clutching function for *E*_{10} arises from the stable homotopy group π_{8}(O(10)) = **Z**/2**Z**.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel-Whitney class *w*_{9} of *E*_{10} vanishes by the Wu formula *w*_{9} = *w*_{1}*w*_{8} + Sq^{1}(*w*_{8}). Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of *E*_{10} with any trivial bundle remains nontrivial. (Hatcher 2009, p. 76)

Given a 2*k*-dimensional vector bundle *E* we have

where *e*(*E*) denotes the Euler class of *E*, and denotes the cup product of cohomology classes.

### Pontryagin classes and curvature

As was shown by Shiing-Shen Chern and André Weil around 1948, the rational Pontryagin classes

can be presented as differential forms which depend polynomially on the curvature form of a vector bundle. This Chern–Weil theory revealed a major connection between algebraic topology and global differential geometry.

For a vector bundle *E* over a *n*-dimensional differentiable manifold *M* equipped with a connection, the total Pontryagin class is expressed as

where Ω denotes the curvature form, and *H**_{dR}(*M*) denotes the de Rham cohomology groups.

### Pontryagin classes of a manifold

The **Pontryagin classes of a smooth manifold** are defined to be the Pontryagin classes of its tangent bundle.

Novikov proved in 1966 that if manifolds are homeomorphic then their rational Pontryagin classes *p _{k}*(

*M*,

**Q**) in

*H*

^{4k}(

*M*,

**Q**) are the same.

If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.

## Pontryagin numbers

**Pontryagin numbers** are certain topological invariants of a smooth manifold. The Pontryagin number vanishes if the dimension of manifold is not divisible by 4. It is defined in terms of the Pontryagin classes of a manifold as follows:

Given a smooth 4*n*-dimensional manifold *M* and a collection of natural numbers

*k*_{1},*k*_{2}, ...,*k*such that_{m}*k*_{1}+*k*_{2}+...+*k*=_{m}*n*.

the Pontryagin number is defined by

where *p _{k}* denotes the

*k*-th Pontryagin class and [

*M*] the fundamental class of

*M*.

### Properties

- Pontryagin numbers are oriented cobordism invariant; and together with Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class.
- Pontryagin numbers of closed Riemannian manifold (as well as Pontryagin classes) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.
- Such invariants as signature and -genus can be expressed through Pontryagin numbers.

## Generalizations

There is also a *quaternionic* Pontryagin class, for vector bundles with quaternion structure.

## See also

## References

- Milnor John W.; Stasheff, James D. (1974).
*Characteristic classes*.*Annals of Mathematics Studies*. Princeton, New Jersey; Tokyo: Princeton University Press / University of Tokyo Press. ISBN 0-691-08122-0. - Hatcher, Allen (2009). "Vector Bundles & K-Theory" (2.1 ed.).

## External links

- Hazewinkel, Michiel, ed. (2001), "Pontryagin class",
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4