Editing Chern class (section)

=== Manifolds with structure ===

The theory of Chern classes gives rise to [[cobordism]] invariants for [[almost complex manifold]]s.

If ''M'' is an almost complex manifold, then its [[tangent bundle]] is a complex vector bundle.  The '''Chern classes''' of ''M'' are thus defined to be the Chern classes of its tangent bundle.  If ''M'' is also [[Compact space|compact]] and of dimension 2''d'', then each [[monomial]] of total degree 2''d'' in the Chern classes can be paired with the [[fundamental class]] of ''M'', giving an integer, a '''Chern number''' of ''M''. If ''M''′ is another almost complex manifold of the same dimension, then it is cobordant to ''M'' if and only if the Chern numbers of ''M''′ coincide with those of ''M''.

The theory also extends to real [[Symplectic geometry|symplectic]] vector bundles, by the intermediation of compatible almost complex structures. In particular, [[symplectic manifold]]s have a well-defined Chern class.