Gysin sequence

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In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given the Euler class of the sphere bundle and vice versa. It was introduced by Werner Gysin in the 1942 article Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten.

1 Definition
- 1.1 De Rham cohomology
- 1.2 Integral cohomology
2 References
3 See also

[edit] Definition

Consider an oriented sphere bundle with total space E, base space M, fiber S^k and projection map

$\pi:E\longrightarrow M.$

Any such bundle defines a degree k+1 cohomology class e called the Euler class of the bundle.

[edit] De Rham cohomology

Discussion of the sequence is most clear in de Rham cohomology. There cohomology classes are represented by differential forms, so that e can be represented by a (k+1)-form.

The projection map π induces a map in cohomology H^* called its pullback π^*

$\pi^*:H^*(M)\longrightarrow H^*(E).$

One can also define a pushforward map π_*

$\pi_*:H^*(E)\longrightarrow H^*(M)$

which acts by fiberwise integration of differential forms on the sphere.

Gysin proved that the following is a long exact sequence $...\longrightarrow H^n(E)\longrightarrow^{\!\!\!\!\!\!\!\!\!\!\pi_*}H^{n-k}(M)\longrightarrow^{\!\!\!\!\!\!\!\!\!\!e\wedge}H^{n+1}(M)\longrightarrow^{\!\!\!\!\!\!\!\!\!\!\pi^*}H^{n+1}(E)\longrightarrow ...$

where $e\wedge$ is the wedge product of a differential form with the Euler class e.

[edit] Integral cohomology

The Gysin sequence is a long exact sequence not only for the de Rham cohomology of differential forms, but also for cohomology with integral coefficients. In the integral case one needs to replace the wedge product with the Euler class with the cup product, and the pushforward map no longer corresponds to integration.