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   over  , its  -th Pontryagin class   is defined as



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

The rational Pontryagin class   is defined to be the image of   in  , the  -cohomology group of   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   and   over  . In terms of the individual Pontryagin classes  ,


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   of   vanishes by the Wu formula  . Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of   with any trivial bundle remains nontrivial. (Hatcher 2009, p. 76)

Given a  -dimensional vector bundle   we have


where   denotes the Euler class of  , 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   over a  -dimensional differentiable manifold   equipped with a connection, the total Pontryagin class is expressed as


where   denotes the curvature form, and   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   in   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


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   vanishes if the dimension of   is not divisible by 4. It is defined in terms of the Pontryagin classes of the manifold   as follows:

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

  such that  ,

the Pontryagin number   is defined by


where   denotes the  -th Pontryagin class and   the fundamental class of  .

Properties edit

  1. Pontryagin numbers are oriented cobordism invariant; and together with Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class.
  2. 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.
  3. 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

  1. ^ "De Rham Cohomology - an overview | ScienceDirect Topics". www.sciencedirect.com. Retrieved 2022-02-02.
  2. ^ Mclean, Mark. "Pontryagin Classes" (PDF). Archived (PDF) from the original on 2016-11-08.
  3. ^ "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. 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 edit