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.
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