Definitions

Pontryagin class

In mathematics, 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.

Definition

Given a vector bundle E over M, its k-th Pontryagin class $p_k\left(E\right)$ is defined as
$p_k\left(E\right)=p_k\left(E,mathbb\left\{Z\right\}\right)=\left(-1\right)^k c_\left\{2k\right\}\left(E otimes mathbb\left\{C\right\}\right)in H^\left\{4k\right\}\left(M,mathbb\left\{Z\right\}\right).$
Here $c_\left\{2k\right\}\left(E otimes mathbb\left\{C\right\}\right)$ denotes the 2k-th Chern class of the complexification $E otimes mathbb\left\{C\right\}=Eoplus i E$ of E and $H^\left\{4k\right\}\left(M,mathbb\left\{Z\right\}\right)$, the 4k-cohomology group of $M$ with integer coefficients.

The rational Pontryagin class $p_k\left(E,\left\{mathbb Q\right\}\right)$ is defined to be image of $p_k\left(E\right)$ in $H^\left\{4k\right\}\left(M,mathbb\left\{Q\right\}\right)$, the 4k-cohomology group of $M$ 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.

Properties

If all Pontryagin classes and Stiefel-Whitney classes of $E$ vanish then the bundle is stably trivial, i.e. its Whitney sum with a trivial bundle is trivial. The total Pontryagin class
$p\left(E\right)=1+p_1\left(E\right)+p_2\left(E\right)+cdotsin H^\left\{*\right\}\left(M,mathbb\left\{Z\right\}\right),$
is multiplicative with respect to Whitney sum of vector bundles, i.e.,
$p\left(Eoplus F\right)=p\left(E\right)cup p\left(F\right)$
for two vector bundles E and F over M, i.e.
$p_1\left(Eoplus F\right)=p_1\left(E\right)+p_1\left(F\right),$
$p_2\left(Eoplus F\right)=p_2\left(E\right)+p_1\left(E\right)cup p_1\left(F\right)+p_2\left(F\right)$
and so on. Given a 2k-dimensional vector bundle E we have
$p_k\left(E\right)=e\left(E\right)cup e\left(E\right),$
where $e\left(E\right)$ denotes the Euler class of E, and $cup$ 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

$p_k\left(E,mathbb\left\{Q\right\}\right)in H^\left\{4k\right\}\left(M,mathbb\left\{Q\right\}\right)$
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

$\left\{rm Tr\right\}\left(Omegawedge cdotswedgeOmega\right)$
constructed with 2k copies of the curvature form $Omega$. In particular the value
$p_k\left(E,mathbb\left\{Q\right\}\right)=\left[\left\{rm Tr\right\}\left(OmegawedgecdotswedgeOmega\right)\right]in H^\left\{4k\right\}_\left\{dR\right\}\left(M\right)$
does not depend on the choice of connection. Here
$H^\left\{*\right\}_\left\{dR\right\}\left(M\right)$
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's theorem states that if manifolds are homeomorphic then their rational Pontryagin classes

$p_k\left(M,mathbb\left\{Q\right\}\right) in H^\left\{4k\right\}\left(M,mathbb\left\{Q\right\}\right)$
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.

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

$k_1,k_2,dots,k_m$ such that $k_1+k_2+cdots+k_m=n$
the Pontryagin number $P_\left\{k_1,k_2,dots,k_m\right\}$ is defined by
$P_\left\{k_1,k_2,dots, k_m\right\}=p_\left\{k_1\right\}cup p_\left\{k_2\right\}cup cdotscup p_\left\{k_m\right\}\left(\left[M\right]\right)$
where $p_\left\{k\right\}$ denotes the k-th Pontryagin class and [M] the fundamental class of M.

Properties

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 manifold (as well as Pontryagin classes) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.
3. Such invariants as signature and $hat A$-genus can be expressed through Pontryagin numbers.

Generalizations

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