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.
De Rham cohomology
The projection map induces a map in cohomology called its pullback
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.
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
- σ is the specialization homomorphism; which sends a k-dimensional subvariety V to the normal cone to the intersection of V and X' in V. The result lies in N through .
- The second map is the (usual) Gysin homomorphism induced by the zero-section embedding .
Example: Given a vector bundle E, let s: X → E be a section of E. Then, when s is a regular section, is the class of the zero-locus of s, where [X] is the fundamental class of X.
- Raoul Bott and Loring Tu, Differential Forms in Algebraic Topology. Springer-Verlag, 1982.
- William Fulton (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1700-8, ISBN 978-3-540-62046-4, MR 1644323
- 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