Gysin homomorphism

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 ${\displaystyle \pi }$ induces a map in cohomology ${\displaystyle H^{\ast }}$ called its pullback ${\displaystyle \pi ^{\ast }}$

${\displaystyle \pi ^{*}:H^{*}(M)\longrightarrow H^{*}(E).\,}$

In the case of a fiber bundle, one can also define a pushforward map ${\displaystyle \pi _{\ast }}$

${\displaystyle \pi _{*}:H^{*}(E)\longrightarrow H^{*-k}(M)}$

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

${\displaystyle \cdots \longrightarrow H^{n}(E){\stackrel {\pi _{*}}{\longrightarrow }}H^{n-k}(M){\stackrel {e_{\wedge }}{\longrightarrow }}H^{n+1}(M){\stackrel {\pi ^{*}}{\longrightarrow }}H^{n+1}(E)\longrightarrow \cdots }$

where ${\displaystyle e_{\wedge }}$ 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.