In
mathematics, the fundamental class is a
homology class
'M''associated to a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
compact manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example i ...
of dimension ''n'', which corresponds to the generator of the homology group
. The fundamental class can be thought of as the orientation of the top-dimensional
simplices
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
of a suitable triangulation of the manifold.In past years mathematics....
Definition
Closed, orientable
When ''M'' is a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
closed manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example ...
of dimension ''n'', the top homology group is
infinite cyclic
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
:
, 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 the direct sum of the fundamental classes for each connected component (corresponding to an orientation for each component).
In relation with
de Rham cohomology it represents ''integration over M''; namely for ''M'' a smooth manifold, an
''n''-form ω can be paired with the fundamental class as
:
which is the integral of ω over ''M'', and depends only on the cohomology class of ω.
Stiefel-Whitney class
If ''M'' is not orientable,
, and so one cannot define a fundamental class ''M'' living inside the integers. However, every closed manifold is
-orientable, and
(for ''M'' connected). Thus every closed manifold is
-oriented (not just orient''able'': there is no ambiguity in choice of orientation), and has a
-fundamental class.
This
-fundamental class is used in defining
Stiefel–Whitney class.
With boundary
If ''M'' is a compact orientable manifold with boundary, then the top relative homology group is again infinite cyclic
, and the notion of the fundamental class is extended to the relative case.
Poincaré duality
For any abelian group
and non negative integer
one can obtain an isomorphism
:
.
using the cap product of the fundamental class and the
-cohomology group . This isomorphism gives Poincaré duality:
:
.
Poincaré duality is extended to the relative case .
See also
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 ...
Applications
In the
Bruhat decomposition In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the principl ...
of the
flag variety of a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
, the fundamental class corresponds to the top-dimension
Schubert cell In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using linea ...
, or equivalently the
longest element of a Coxeter group.
See also
*
Longest element of a Coxeter group
*
Poincaré duality
References
*
External links
Fundamental classat the Manifold Atlas.
* The Encyclopedia of Mathematics article o
the fundamental class
{{DEFAULTSORT:Fundamental Class
Algebraic topology