Gysin sequence

Definition

Consider a fiber-oriented sphere bundle with total space E, base space M, fiber S^k and projection map π : E ⟶ M . 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 π induces a map in cohomology H^* called its pullback π^*

In the case of a fiber bundle, one can also define a pushforward map π_*

which acts by fiberwise integration of differential forms on the sphere (cf. integration along fibers) – 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 e ∧ 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.

Related concepts

The Gysin map, π ∗ : H ∗ ( E ) ⟶ H ∗ ( M ) , is a covariant map between objects associated with a contravariant functor – it goes "the wrong way". Other such maps are called "wrong way maps", Gysin maps – because of their occurrence in this sequence – or other terms such as shriek maps or "transfer maps".