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The Pontryagin number is defined in terms of the Pontryagin Class of a Manifold as follows. For any collection of Pontryagin Classes such that their cup product has the same Dimension as the Manifold, this cup product can be evaluated on the Manifold's Fundamental Class. The resulting number is called the Pontryagin number for that combination of Pontryagin classes. The most important aspect of Pontryagin numbers is that they are Cobordism invariant. Together, Pontryagin and Stiefel-Whitney Numbers determine an oriented manifold's oriented Cobordism class.
See also Chern Number, Stiefel-Whitney Number