Euler class
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 E → X is an oriented, real vector bundle of rank r.
The Euler class e(E) is an element of the integral cohomology group
constructed as follows. An orientation of E amounts to a continuous choice of generator of the cohomology
of each fiber F relative to the complement F\F0 of its zero element F0. From Thom isomorphism, this induces an orientation class
in the cohomology of E relative to the complement E\E0 of the zero section E0. The inclusions
where X includes into E 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:
Note that "Normalization" is a distinguishing feature of the Euler class, so that it detects the existence of a non-vanishing section
Also unlike other characteristic classes, it is concentrated in a single dimension, which depends on the rank of the bundle: e(E) ∈ Hr — there are no e0, e1, .... In particular, c0(E) = p0(E) = 1 ∈ H0(X; Z) and w0(E) = 1 ∈ H0(X; Z/2Z), but there is no e0. This reflects the fact that the Euler class is unstable, as discussed below.
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