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
:
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
induces a map in cohomology
called its
pullback
:
In the case of a fiber bundle, one can also define a
pushforward map
:
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
:
where
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
:
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
.
* The second map is the (usual) Gysin homomorphism induced by the zero-section embedding
.
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
Example: Given a vector bundle ''E'', let ''s'': ''X'' → ''E'' be a section of ''E''. Then, when ''s'' is a
regular section,