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.
|[a1]||E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 156|
Euler class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_class&oldid=31830