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.
is defined as where: The rational Pontryagin class
The total Pontryagin class is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e., for two vector bundles
In terms of the individual Pontryagin classes
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
arises from the homotopy group
vanishes by the Wu formula
Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of
with any trivial bundle remains nontrivial.
denotes the cup product of cohomology classes.
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.
equipped with a connection, the total Pontryagin class is expressed as where
denotes the curvature form, and
denotes the de Rham cohomology groups.
[citation needed] 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
If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
The Pontryagin classes of a complex vector bundle
is completely determined by its Chern classes.
, the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle.
[2]for example, we can apply this formula to find the Pontryagin classes of a complex vector bundle on a curve and a surface.
so all of the Pontryagin classes of complex vector bundles are trivial.
Recall that a quartic polynomial whose vanishing locus in
corresponds to four points, due to Bézout's lemma, we have the second chern number as
This number can be used to compute the third stable homotopy group of spheres.
[3] Pontryagin numbers are certain topological invariants of a smooth manifold.
It is defined in terms of the Pontryagin classes of the manifold