# Pontryagin class

In mathematics, the **Pontryagin classes**, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.

## Definition[edit]

Given a real vector bundle *E* over *M*, its *k*-th Pontryagin class is defined as

where:

- denotes the -th Chern class of the complexification of
*E*, - is the -cohomology group of
*M*with integer coefficients.

The rational Pontryagin class is defined to be the image of in , the -cohomology group of *M* with rational coefficients.

## Properties[edit]

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 over the 9-sphere. (The clutching function for arises from the homotopy group .) 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[edit]

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.^{[1]}

### Pontryagin classes of a manifold[edit]

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

Novikov proved in 1966 that if two compact, oriented, smooth 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 classes from Chern classes[edit]

The Pontryagin classes of a complex vector bundle is completely determined by its Chern classes. This follows from the fact that , the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is, and . Then, this given the relation

^{[2]}

for example, we can apply this formula to find the Pontryagin classes of a complex vector bundle on a curve and a surface. For a curve, we have

so all of the Pontryagin classes of complex vector bundles are trivial. On a surface, we have

showing . On line bundles this simplifies further since by dimension reasons.

### Pontryagin classes on a Quartic K3 Surface[edit]

Recall that a quartic polynomial whose vanishing locus in is a smooth subvariety is a K3 surface. If we use the normal sequence

we can find

showing and . Since corresponds to four points, due to Bezout's lemma, we have the second chern number as . Since in this case, we have

. This number can be used to compute the third stable homotopy group of spheres.^{[3]}

## Pontryagin numbers[edit]

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

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

- such that ,

the Pontryagin number is defined by

where denotes the *k*-th Pontryagin class and [*M*] the fundamental class of *M*.

### Properties[edit]

- 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 manifolds (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold.
- Invariants such as signature and -genus can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see Hirzebruch signature theorem.

## Generalizations[edit]

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

## See also[edit]

## References[edit]

**^**"De Rham Cohomology - an overview | ScienceDirect Topics".*www.sciencedirect.com*. Retrieved 2022-02-02.**^**Mclean, Mark. "Pontryagin Classes" (PDF). Archived (PDF) from the original on 2016-11-08.**^**"A Survey of Computations of Homotopy Groups of Spheres and Cobordisms" (PDF). p. 16. Archived (PDF) from the original on 2016-01-22.

- Milnor John W.; Stasheff, James D. (1974).
*Characteristic classes*. Princeton, New Jersey; Tokyo: Princeton University Press / University of Tokyo Press. ISBN 0-691-08122-0.`{{cite book}}`

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ignored (help) - Hatcher, Allen (2009). "Vector Bundles & K-Theory" (2.1 ed.).
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## External links[edit]

- "Pontryagin class",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]