The Ultimate Pontryagin number - American History Information Guide and Reference

In differential topology, 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,... k m

such that

k 1 + k 2 + ... + k m = n

the Pontryagin number $P_{k_1,k_2,...k_m}$ is defined by

$P_{k_1,k_2,...k_m}=p_{k_1}\cup p_{k_2}\cup ...\cup p_{k_m}([M])$

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

Properties

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 $\hat A$ -genus can be expressed through Pontryagin numbers.