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 , 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
formula_1:
formula_2
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 formula_1 induces a map in cohomology formula_4 called its pullback formula_5
In the case of a fiber bundle, one can also define a pushforward map formula_7
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 formula_10 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.
Gysin homomorphism in algebraic geometry.
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. Let "N" be the pullback of the normal bundle of "i" to "X". Then the refined Gysin homomorphism "i"! refers to the composition
where
The homomorphism "i"! "encodes" intersection product in intersection theory in that one either shows or defines the intersection product of "X" and "V" as: formula_14
Example: Given a vector bundle "E", let "s": "X" → "E" be a section of "E". Then, when "s" is a regular section, formula_15 is the class of the zero-locus of "s", where ["X"] is the fundamental class of "X".