Characteristic Class - Characteristic Numbers

Characteristic Numbers

For characteristic numbers in fluid dynamics, see characteristic number (fluid dynamics).

Characteristic classes are elements of cohomology groups; one can obtain integers from characteristic classes, called characteristic numbers. Respectively: Stiefel-Whitney numbers, Chern numbers, Pontryagin numbers, and the Euler characteristic.

Given an oriented manifold M of dimension n with fundamental class, and a G-bundle with characteristic classes, one can pair a product of characteristic classes of total degree n with the fundamental class. The number of distinct characteristic numbers is the number of monomials of degree n in the characteristic classes, or equivalently the partitions of n into .

Formally, given such that, the corresponding characteristic number is:

These are notated various as either the product of characteristic classes, such as or by some alternative notation, such as for the Pontryagin number corresponding to, or for the Euler characteristic.

From the point of view of de Rham cohomology, one can take differential forms representing the characteristic classes, take a wedge product so that one obtains a top dimensional form, then integrates over the manifold; this is analogous to taking the product in cohomology and pairing with the fundamental class.

This also works for non-orientable manifolds, which have a -orientation, in which case one obtains -valued characteristic numbers, such as the Stiefel-Whitney numbers.

Characteristic numbers solve the oriented and unoriented bordism questions: two manifolds are (respectively oriented or unoriented) cobordant if and only if their characteristic numbers are equal.