Euler class

This article defines a characteristic class

Definition

The Euler class is a characteristic class for the topological group $GL^{+}(n,\mathbb {R} )$ 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.