, the Pontryagin classes
are certain characteristic classes
. The Pontryagin class lies in cohomology groups
with index a multiple of four. It applies to real vector bundles
Given a vector bundle E
, its k
-th Pontryagin class
is defined as
denotes the 2k
-th Chern class
of the complexification
, the 4k
The rational Pontryagin class is defined to be image of in , the 4k-cohomology group of with rational coefficients.
Pontryagin classes have a meaning in real differential geometry — unlike the Chern class, which assumes a complex vector bundle at the outset.
If all Pontryagin classes and Stiefel-Whitney classes
vanish then the bundle is stably trivial,
i.e. its Whitney sum
with a trivial bundle is trivial.
The total Pontryagin class
is multiplicative with respect to Whitney sum
of vector bundles, i.e.,
for two vector bundles E
and so on.
Given a 2k
-dimensional vector bundle E
denotes the Euler class
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, its k-th Pontryagin class can be realized by the 4k-form
constructed with 2k
copies of the curvature form
. In particular the value
does not depend on the choice of connection. Here
denotes the de Rham cohomology
Pontryagin classes of a manifold
The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle.
Novikov's theorem states that if manifolds are homeomorphic then their rational Pontryagin classes
are the same.
If the dimension is at least five, there at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
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
Given a smooth 4n-dimensional manifold M and a collection of natural numbers
- such that
the Pontryagin number
is defined by
denotes the k
-th Pontryagin class and [M
] the fundamental class
- 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.
There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.