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 ${\displaystyle E}$ is an oriented, real vector bundle of rank ${\displaystyle r}$ over a base space ${\displaystyle X}$.