Gysin sequence

The Gysin sequence is in mathematics, specifically in algebraic topology, a long exact sequence relating the Kohomologieklassen of base, fiber and total space of a bundle spheres relate to each other. An application to calculate the cohomology of the Euler class (and vice versa ) of a sphere bundle dar.

The sequence was introduced in 1942 by Werner Gysin.

Definition

Let E be an oriented sphere bundle, M is the associated base, Sk the typical fiber and the projection map. Such a bundle can be a cohomology assign e of degree k 1, is called the Euler class of the bundle.

The projection map on the base induces a map in cohomology, the so-called pullback. Furthermore, there is a " Push Forward " mentioned homomorphism.

Gysin showed that the following long sequence is exactly:

The easiest way to the sequence in de Rham cohomology describe. Here Kohomologieklassen given by differential shapes Euler class can therefore be represented by a k 1 form. The push -forward mapping is given by fiber- wise integration of differential forms on the sphere and in the sequence denotes the exterior product of differential forms. In contrast, integral cohomology can to push forward no longer regarded as integration and the wedge product must be replaced by the cup-product.