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. 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. There cohomology classes are represented by differential forms, so that e can be represented by a (k + 1)-form. 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. 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 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 .