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Concept
Euler class
Summary
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this.
Throughout this article is an oriented, real vector bundle of rank over a base space .
The Euler class is an element of the integral cohomology group
constructed as follows. An orientation of amounts to a continuous choice of generator of the cohomology
of each fiber relative to the complement of zero. From the Thom isomorphism, this induces an orientation class
in the cohomology of relative to the complement of the zero section . The inclusions
where includes into as the zero section, induce maps
The Euler class e(E) is the image of u under the composition of these maps.
The Euler class satisfies these properties, which are axioms of a characteristic class:
Functoriality: If is another oriented, real vector bundle and is continuous and covered by an orientation-preserving map , then . In particular, .
Whitney sum formula: If is another oriented, real vector bundle, then the Euler class of their direct sum is given by
Normalization: If possesses a nowhere-zero section, then .
Orientation: If is with the opposite orientation, then .
Note that "Normalization" is a distinguishing feature of the Euler class. The Euler class obstructs the existence of a non-vanishing section in the sense that if then has no non-vanishing section.
Also unlike other characteristic classes, it is concentrated in a degree which depends on the rank of the bundle: . By contrast, the Stiefel Whitney classes live in independent of the rank of . This reflects the fact that the Euler class is unstable, as discussed below.
The Euler class corresponds to the vanishing locus of a section of in the following way. Suppose that is an oriented smooth manifold of dimension .
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