Gysin sequence

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 Gysin (1942), and is generalized by the Serre spectral sequence.

Definition

Consider a fiber-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.

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). \,

In the case of a fiber bundle, 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 (cf. integration along fibers) – note that this map goes "the wrong way": it is a covariant map between objects associated with a contravariant functor.

Gysin proved that the following is a long exact sequence

\dots \longrightarrow H^n(E) \stackrel{\pi_*}{\longrightarrow} H^{n-k}(M) \stackrel{e_\wedge}{\longrightarrow} H^{n+1}(M) \stackrel{\pi^*}{\longrightarrow} H^{n+1}(E) \longrightarrow \dots

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

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.

Related concepts

For more details on this topic, see Shriek map.

The Gysin map, $\pi_*\colon H^*(E) \longrightarrow H^*(M),$ is a covariant map between objects associated with a contravariant functor – it goes "the wrong way". Other such maps are called "wrong way maps", Gysin maps – because of their occurrence in this sequence – or other terms such as shriek maps or "transfer maps".

References

Raoul Bott and Loring Tu, Differential Forms in Algebraic Topology. Springer-Verlag, 1982.
Gysin, Werner (1942), "Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten", Commentarii Mathematici Helvetici 14: 61–122, doi:10.1007/bf02565612, ISSN 0010-2571, MR 0006511