Euler class

Summary

The Euler class $e(E) \in H^n(B; \Z)$ is a characteristic class of oriented real vector bundles $E \to B$ of rank $n$. It is the first obstruction to a nowhere-zero section: if $E$ has a section that never vanishes, then $e(E) = 0$. For the tangent bundle of a closed oriented manifold, its value on the fundamental class is the Euler characteristic. It appears in the Gysin sequence and is defined from the Thom class of the Thom isomorphism.