Editing Pontryagin class (section)

== Pontryagin numbers ==
'''Pontryagin numbers''' are certain [[topological invariant]]s of a smooth [[manifold]]. Each Pontryagin number of a manifold <math>M</math> vanishes if the dimension of <math>M</math> is not divisible by 4. It is defined in terms of the Pontryagin classes of the [[manifold]] <math>M</math> as follows:

Given a smooth <math>4 n</math>-dimensional manifold <math>M</math> and a collection of natural numbers 
:<math>k_1, k_2, \ldots , k_m</math> such that <math>k_1+k_2+\cdots +k_m =n</math>,
the Pontryagin number <math>P_{k_1,k_2,\dots,k_m}</math> is defined by
:<math>P_{k_1,k_2,\dots, k_m}=p_{k_1}\smile p_{k_2}\smile \cdots\smile p_{k_m}([M])</math>
where <math>p_k</math> denotes the <math>k</math>-th Pontryagin class and <math>[M]</math> the [[fundamental class]] of <math>M</math>.

=== Properties ===
#Pontryagin numbers are oriented [[cobordism]] invariant; and together with [[Stiefel-Whitney number]]s they determine an oriented manifold's oriented cobordism class.
#Pontryagin numbers of closed [[Riemannian manifold]]s (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold.
#Invariants such as [[Signature (topology)|signature]] and [[Â genus|<math>\hat A</math>-genus]] can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see [[Hirzebruch signature theorem]].