Euler class

This article defines a characteristic class

Definition

The Euler class is a characteristic class for the topological group $G{L}^{+}(n,\mathbb{R})$ with coefficients in $\mathbb{Z}$.

Given a real vector bundle $p:E\to B$ of dimension $n$, the Euler class of $E$, denoted $e(E)$, is an element of $H^n(B;\mathbb{Z})$ obtained as the restriction of a Thom class $c\in H^n(D(E),S(E);\mathbb{Z})$ under the composition:

$H^n(D(E),S(E);\mathbb{Z})\to H^n(D(E);\mathbb{Z})\to H^n(B;\mathbb{Z})$

The first map is induced by inclusion and the second map by inclusion as the zero section.