Euler Class - Relations To Other Invariants

Relations To Other Invariants

In the special case when the bundle in question is the tangent bundle of a compact, oriented, -dimensional manifold, the Euler class is an element of the top cohomology of the manifold, which is naturally identified with the integers by evaluating cohomology classes on the fundamental homology class. Under this identification, the Euler class of the tangent bundle equals the Euler characteristic of the manifold. In the language of characteristic numbers, the Euler characteristic is the characteristic number corresponding to the Euler class.

Thus the Euler class is a generalization of the Euler characteristic to vector bundles other than tangent bundles. In turn, the Euler class is the archetype for other characteristic classes of vector bundles, in that each "top" characteristic class equals the Euler class, as follows.

Modding out by induces a map

The image of the Euler class under this map is the top Stiefel-Whitney class . One can view this Stiefel-Whitney class as "the Euler class, ignoring orientation".

Any complex vector bundle of complex rank can be regarded as an oriented, real vector bundle of real rank . The top Chern class of the complex bundle equals the Euler class of the real bundle.

The Whitney sum is isomorphic to the complexification, which is a complex bundle of rank . Comparing Euler classes, we see that