Pontryagin number

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 4 n -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 (topology) and hat A-genus can be expressed through Pontryagin numbers.