Gysine sequence

In mathematics , more precisely in algebraic topology , the Gysin sequence is a long, exact sequence that relates the cohomology classes of base, fiber and total space of a bundle of spheres . One application is the calculation of the cohomology from the Euler class (and vice versa) of a bundle of spheres.

The sequence was introduced in 1942 by Werner Gysin .

definition

Let be an oriented bundle of spheres, the associated base, the typical fiber and the projection image. Such a bundle can be assigned a cohomology class of degree , which is called the Euler class of the bundle. ${\displaystyle E}$ ${\displaystyle M}$ ${\displaystyle S^{k}}$ ${\displaystyle \pi :E\longrightarrow M}$ ${\displaystyle e}$ ${\displaystyle k+1}$

The projection mapping onto the base induces mapping in the cohomology , the so-called pullback . There is also a homomorphism called “push forward” . ${\displaystyle \pi }$ ${\displaystyle H^{*}}$ ${\displaystyle \pi ^{*}:H^{*}(M)\longrightarrow H^{*}(E)}$ ${\displaystyle \pi _{*}:H^{n}(E)\rightarrow H^{n-k}(M)}$

Gysin showed that the following long sequence is exact: ${\displaystyle \ldots \longrightarrow H^{n}(E)\longrightarrow ^{\!\!\!\!\!\!\!\pi _{*}}H^{n-k}(M)\longrightarrow ^{\!\!\!\!\!\!\!e\wedge }H^{n+1}(M)\longrightarrow ^{\!\!\!\!\!\!\!\pi ^{*}}H^{n+1}(E)\longrightarrow \ldots }$

The easiest way to describe the sequence is in De Rham cohomology . Here the cohomology classes are given by differential forms , so the Euler class can be represented by a -form. The pushforward mapping is given by the fiber-wise integration of differential forms on the sphere and in the sequence denotes the outer product of differential forms. In integral cohomology, on the other hand, the push forward can no longer be understood as integration and the wedge product must be replaced by the cup product . ${\displaystyle k+1}$ ${\displaystyle \wedge }$

literature

• Werner Gysin: "On the homology theory of the mapping and fiberization of manifolds", Commentarii Mathematici Helvetici, 14, 61–122 (1942). online (PDF)
• Raoul Bott , Loring W. Tu: Differential Forms in Algebraic Topology. 4th printing. Springer, New York et al. 2008, ISBN 978-0-387-90613-3 ( Graduate Texts in Mathematics 82).

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