Editing Chern class (section)

===Chern numbers===

If we work on an [[orientable manifold|oriented manifold]] of dimension <math>2n</math>, then any product of Chern classes of total degree <math>2n</math> (i.e., the sum of indices of the Chern classes in the product should be <math>n</math>) can be paired with the [[orientation homology class]] (or "integrated over the manifold") to give an integer, a '''Chern number''' of the vector bundle. For example, if the manifold has dimension 6, there are three linearly independent Chern numbers, given by <math>c_1^3</math>, <math>c_1 c_2</math>, and <math>c_3</math>.  In general, if the manifold has dimension <math>2n</math>, the number of possible independent Chern numbers is the number of [[integer partition|partition]]s of <math>n</math>.

The Chern numbers of the tangent bundle of a complex (or almost complex) manifold are called the Chern numbers of the manifold, and are important invariants.