Euler 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.