Gysin sequence
Encyclopedia
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 sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings
given the Euler class
of the sphere bundle and vice versa. It was introduced by Werner Gysin in the 1942 article Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten, and is generalized by the Serre spectral sequence
.
Any such bundle defines a degree k+1 cohomology class e called the Euler class of the bundle.
. Their cohomology classes are represented by differential forms
, so that e can be represented by a (k+1)-form.
The projection map π induces a map in cohomology H* called its pullback
π*
In the case of a fiber bundle, one can also define a pushforward
map π*
which acts by fiberwise integration of differential forms on the 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.
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.
is a covariant map between objects associated with a contravariant 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".
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
known as algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although 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 sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings
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 understood as singular cohomology, but the ring structure is...
given the Euler class
Euler class
In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...
of the sphere bundle and vice versa. It was introduced by Werner Gysin in the 1942 article Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten, and is generalized by the Serre spectral sequence
Serre spectral sequence
In mathematics, the Serre spectral sequence is an important tool in algebraic topology...
.
Definition
Consider a fiber-oriented sphere bundle with total space E, base space M, fiber Sk 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 most clear in de Rham cohomologyDe Rham cohomology
In mathematics, de Rham cohomology 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 adapted to computation and the concrete representation of cohomology classes...
. Their cohomology classes are represented by differential forms
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
, so that e can be represented by a (k+1)-form.
The projection map π induces a map in cohomology H* called its pullback
Pullback
Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback , and is frequently denoted by φ*...
π*
In the case of a fiber bundle, one can also define a pushforward
Pushforward
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different, but closely related things.*Pushforward : the differential of a smooth map between manifolds, and the "pushforward" operations it defines.**Pushforward #Pushforward of vector...
map π*
which acts by fiberwise integration of differential forms on the sphere – note that this map goes "the wrong way"
Shriek map
In category theory, a branch of mathematics, certain unusual functors are denoted f_! and f^!, with the exclamation mark used to indicate that they are exceptional in some 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 cohomologyDe Rham cohomology
In mathematics, de Rham cohomology 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 adapted to computation and the concrete representation of cohomology classes...
of differential forms, but also for cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
with integral coefficients. In the integral case one needs to replace the wedge product with the Euler class with the cup product
Cup product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...
, and the pushforward map no longer corresponds to integration.
Related concepts
The Gysin map, is a covariant map between objects associated with a contravariant 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 mapShriek map
In category theory, a branch of mathematics, certain unusual functors are denoted f_! and f^!, with the exclamation mark used to indicate that they are exceptional in some way...
s or "transfer maps".