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In the field of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
known as algebraic topology, the Gysin sequence is a long exact sequence which relates the
cohomology classes In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
of the
base space In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
, the fiber and the
total space In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
of a sphere bundle. The Gysin sequence is a useful tool for calculating the
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually und ...
s given the Euler class of the sphere bundle and vice versa. It was introduced by , and is generalized by the Serre spectral sequence.


Definition

Consider a fiber-oriented sphere bundle with total space ''E'', base space ''M'', fiber ''S''''k'' and projection map \pi: S^k \hookrightarrow E \stackrel 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 clearest with
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
. There cohomology classes are represented by
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s, so that ''e'' can be represented by a (''k'' + 1)-form. The projection map \pi induces a map in cohomology H^\ast called its pullback \pi^\ast :\pi^*:H^*(M)\longrightarrow H^*(E). \, In the case of a fiber bundle, one can also define a pushforward map \pi_\ast :\pi_*:H^*(E)\longrightarrow H^(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 :\cdots \longrightarrow H^n(E) \stackrel H^(M) \stackrel H^(M) \stackrel H^(E) \longrightarrow \cdots where e_\wedge 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 In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
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.


Gysin homomorphism in algebraic geometry

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 :i^!: A_k(Y') \overset\longrightarrow A_k(N) \overset \longrightarrow A_(X') 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 C_ \hookrightarrow N. * The second map is the (usual) Gysin homomorphism induced by the zero-section embedding X' \hookrightarrow N. The homomorphism ''i''! ''encodes''
intersection product In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
in intersection theory in that one either shows, or defines the intersection product of ''X'' and ''V'' by, the formula X \cdot V = i^! Example: Given a vector bundle ''E'', let ''s'': ''X'' → ''E'' be a section of ''E''. Then, when ''s'' is a regular section, s^ /math> is the class of the zero-locus of ''s'', where 'X''is the fundamental class of ''X''.


See also

* Logarithmic form * Wang sequence


Notes


Sources

* * * {{DEFAULTSORT:Gysin Sequence Algebraic topology