Gysin homomorphism

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 S^k and projection map ${\displaystyle \pi :E\longrightarrow M.}$ Any such bundle defines a degree k + 1 cohomology class e called the Euler class of the bundle.

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^*

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