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.
Consider a fiber-oriented sphere bundle with total space E, base space M, fiber Sk and projection map
Any such bundle defines a degree k + 1 cohomology class e called the Euler class of the bundle.
Discussion of the sequence is clearest with de Rham cohomology.
In the case of a fiber bundle, one can also define a pushforward map
which acts by fiberwise integration of differential forms on the oriented sphere – 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 where
is the wedge product of a differential form with the Euler class e. 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.
Let i: X → Y be a (closed) regular embedding of codimension d, Y' → Y a morphism and i': X' = X ×Y Y' → Y' the induced map.