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| 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 [[Fiber_bundle#Sphere_bundles|sphere bundle]]. The Gysin sequence is a useful tool for calculating the [[cohomology ring]]s given the [[Euler class]] of the sphere bundle and vice versa. It was introduced by {{harvs|txt|authorlink=Werner Gysin|last=Gysin|year= 1942}}, and is generalized by the [[Serre spectral sequence]].
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| ==Definition==
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| Consider a fiber-oriented sphere bundle with total space ''E'', base space ''M'', fiber ''S''<sup>''k''</sup> and [[projection map]]
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| :::::<math>\pi:E\longrightarrow M. </math>
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| Any such bundle defines a degree ''k'' + 1 cohomology class ''e'' called the Euler class of the bundle.
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| ===De Rham cohomology===
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| Discussion of the sequence is most clear in [[de Rham cohomology]]. Their cohomology classes are represented by [[differential form]]s, so that ''e'' can be represented by a (''k'' + 1)-form.
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| The projection map π induces a map in cohomology H<sup>*</sup> called its [[pullback]] π<sup>*</sup>
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| ::::<math>\pi^*:H^*(M)\longrightarrow H^*(E). \, </math>
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| In the case of a fiber bundle, one can also define a [[pushforward]] map π<sub>*</sub>
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| ::::<math>\pi_*:H^*(E)\longrightarrow H^*(M) </math>
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| which acts by fiberwise [[integration (mathematics)|integration]] of differential forms on the sphere (cf. [[integration in fiber]]) – note that [[shriek map|this map goes "the wrong way"]]: it is a covariant map between objects associated with a contravariant functor.
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| Gysin proved that the following is a long exact sequence
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| <math>...\longrightarrow H^n(E)\longrightarrow^{\!\!\!\!\!\!\!\!\!\!\pi_*}H^{n-k}(M)\longrightarrow^{\!\!\!\!\!\!\!\!\!\!e\wedge}H^{n+1}(M)\longrightarrow^{\!\!\!\!\!\!\!\!\!\!\pi^*}H^{n+1}(E)\longrightarrow ...</math>
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| where <math>e\wedge</math> is the [[wedge product]] of a differential form with the Euler class ''e''.
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| ===Integral cohomology===
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| 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. | |
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| == Related concepts ==
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| {{details|Shriek map}}
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| The Gysin map, <math>\pi_*\colon H^*(E)\longrightarrow H^*(M),</math> is a ''co''variant map between objects associated with a ''contra''variant 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 map]]s or "transfer maps".
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| ==References==
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| *[[Raoul Bott]] and Loring Tu, ''Differential Forms in Algebraic Topology.'' Springer-Verlag, 1982.
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| *{{Citation | last1=Gysin | first1=Werner | title=Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002053314 | id={{MR|0006511}} | year=1942 | journal=Commentarii Mathematici Helvetici | issn=0010-2571 | volume=14 | pages=61–122}}
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| ==See also==
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| * [[Serre spectral sequence]], a generalization
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| * [[Logarithmic form]]
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| {{DEFAULTSORT:Gysin Sequence}}
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| [[Category:Algebraic topology]]
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The name of the author is Jayson. My wife and I live in Kentucky. Invoicing is my profession. The preferred hobby for him and his children is to perform lacross and he would by no means give it up.
My webpage ... free psychic reading