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 ${\displaystyle X}$.

The Euler class ${\displaystyle e(E)}$ is an element of the integral cohomology group

constructed as follows. An orientation of ${\displaystyle E}$ amounts to a continuous choice of generator of the cohomology

of each fiber ${\displaystyle \mathbf {R} ^{r}}$ relative to the complement ${\displaystyle \mathbf {R} ^{r}\setminus \{0\}}$ of zero. From the Thom isomorphism, this induces an orientation class

in the cohomology of ${\displaystyle E}$ relative to the complement ${\displaystyle E\setminus E_{0}}$ of the zero section ${\displaystyle E_{0}}$. The inclusions

where ${\displaystyle X}$ includes into ${\displaystyle E}$ as the zero section, induce maps

The Euler class e(E) is the image of u under the composition of these maps.