Euler class

The first obstruction in the construction of sections of a fibration the fibre of which is a sphere, associated with a vector bundle. See Characteristic class.

Multiplication by the Euler class (sometimes called the orientation class) is one of the homeomorphisms in the Gysin sequence of an oriented sphere bundle [a1]. The Euler characteristic of a compact, oriented $n$-dimensional manifold may be calculated from the Euler class of the tangent bundle [a1], p. 348.