- Gysin sequence
In the field of
mathematics known asalgebraic topology , theGysin sequence is along exact sequence which relates thecohomology classes of thebase space , the fiber and thetotal space of a sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given theEuler class of the sphere bundle and vice versa. It was introduced byWerner Gysin in the 1942 article [http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D209961 Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten] .Definition
Consider a fiber-oriented sphere bundle with total space "E", base space "M", fiber "S"k and
projection map :::::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 bydifferential form s, 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*::::One can also define apushforward map π*::::which acts by fiberwise integration of differential forms on the sphere.Gysin proved that the following is a long exact sequence
where 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 thecup product , and the pushforward map no longer corresponds to integration.References
*
Raoul Bott and Loring Tu, "Differential Forms in Algebraic Topology." Springer-Verlag, 1982.ee also
*
Serre spectral sequence , a generalization
Wikimedia Foundation. 2010.