Pontryagin Class - 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

such that

the Pontryagin number is defined by

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