Fundamental class
Encyclopedia
In mathematics
, the fundamental class is a homology
class [M] associated to an oriented manifold
M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold". Intuitively, the fundamental class can be thought of as the sum of the (top-dimensional) simplices of a suitable triangulation of the manifold.
orientable closed manifold
of dimension n, the top homology group is infinite cyclic:
, and an orientation is a choice of generator, a choice of isomorphism 
. The generator is called the fundamental class.
If M is disconnected (but still orientable), a fundamental class is a fundamental class for each connected component (corresponding to an orientation for each component).
It represents, in a sense, integration over M, and in relation with de Rham cohomology
it is exactly that; namely for M a smooth manifold, an n-form
ω can be paired with the fundamental class as
to get a real number, which is the integral of ω over M, and depends only on the cohomology class of ω.
(or over 
), as 
(if M is connected), and indeed, one cannot integrate differential n-forms over non-orientable manifolds.
However, every closed manifold is
-orientable, and
(for M connected). Thus every closed manifold is 
-oriented (not just orientable: there is no ambiguity in choice of orientation), and has a 
-fundamental class.
This
-fundamental class is used in defining Stiefel–Whitney numbers.
, and as with closed manifolds, a choice of isomorphism is a fundamental class.
, the fundamental class is dual to the bottom class of a connected manifold (a generator of
): in the closed case, Poincaré duality is the statement that the cap product
with the fundamental class yields an isomorphism
.
See also Twisted Poincaré duality
of the flag variety of a Lie group
, the fundamental class corresponds to the top-dimension Schubert cell, or equivalently the longest element of a Coxeter group
.
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...
, the fundamental class is a homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
class [M] associated to an oriented manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold". Intuitively, the fundamental class can be thought of as the sum of the (top-dimensional) simplices of a suitable triangulation of the manifold.
Closed, orientable
When M is a connectedConnected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
orientable closed manifold
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
of dimension n, the top homology group is infinite cyclic:
, and an orientation is a choice of generator, a choice of isomorphism . The generator is called the fundamental class.If M is disconnected (but still orientable), a fundamental class is a fundamental class for each connected component (corresponding to an orientation for each component).
It represents, in a sense, integration over M, and in relation with de Rham cohomology
De 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...
it is exactly that; namely for M a smooth manifold, an n-form
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...
ω can be paired with the fundamental class as
to get a real number, which is the integral of ω over M, and depends only on the cohomology class of ω.
Non-orientable
If M is not orientable, one cannot define a fundamental class, or more precisely, one cannot define a fundamental class over (or over ), as (if M is connected), and indeed, one cannot integrate differential n-forms over non-orientable manifolds.However, every closed manifold is
-orientable, and (for M connected). Thus every closed manifold is -oriented (not just orientable: there is no ambiguity in choice of orientation), and has a -fundamental class.This
-fundamental class is used in defining Stiefel–Whitney numbers.With boundary
If M is a compact orientable manifold with boundary, then the top relative homology group is again infinite cyclic, and as with closed manifolds, a choice of isomorphism is a fundamental class.Poincaré duality
Under Poincaré dualityPoincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
, the fundamental class is dual to the bottom class of a connected manifold (a generator of
): in the closed case, Poincaré duality is the statement that the cap productCap product
In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938.-Definition:Let X be a topological...
with the fundamental class yields an isomorphism
.See also Twisted Poincaré duality
Twisted Poincaré duality
In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system....
Applications
In the Bruhat decompositionBruhat decomposition
In mathematics, the Bruhat decomposition G = BWB into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases...
of the flag variety of a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
, the fundamental class corresponds to the top-dimension Schubert cell, or equivalently the longest element of a Coxeter group
Longest element of a Coxeter group
In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by w0...
.
See also
- Longest element of a Coxeter groupLongest element of a Coxeter groupIn mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by w0...
- Poincaré dualityPoincaré dualityIn mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...