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{{For|the fundamental class in class field theory|class formation}}
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{{Unreferenced|date=December 2009}}
 
In [[mathematics]], the '''fundamental class''' is a [[homology (mathematics)|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.
 
==Definition==
===Closed, orientable===
When ''M'' is a [[connected space|connected]] [[orientable]] [[closed manifold]] of dimension ''n'', the top homology group is [[infinite cyclic]]: <math>H_n(M,\mathbf{Z}) \cong \mathbf{Z}</math>, and an orientation is a choice of generator, a choice of isomorphism <math>\mathbf{Z} \to H_n(M,\mathbf{Z})</math>. 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 [[differential form|''n''-form]] ω can be paired with the fundamental class as
 
:<math>\langle\omega, [M]\rangle = \int_M \omega</math>
 
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 <math>\mathbf{Z}</math>'' (or over <math>\mathbf{R}</math>), as <math>H_n(M;\mathbf{Z})=0</math>, and indeed, one cannot integrate differential ''n''-forms over non-orientable manifolds.
 
However, every closed manifold is <math>\mathbf{Z}/2</math>-orientable, and  
<math>H_n(M;\mathbf{Z}/2)=\mathbf{Z}/2</math> (for ''M'' connected). Thus every closed manifold is <math>\mathbf{Z}/2</math>-oriented (not just orient''able'': there is no ambiguity in choice of orientation), and has a <math>\mathbf{Z}/2</math>-fundamental class.
 
This <math>\mathbf{Z}/2</math>-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 <math>H_n(M,\partial M)\cong \mathbf{Z}</math>, and as with closed manifolds, a choice of isomorphism is a fundamental class.
 
==Poincaré duality==
{{Main|Poincaré duality}}
{{Expand section|date=December 2008}}
Under [[Poincaré duality]], the fundamental class is dual to the bottom class of a connected manifold (a generator of <math>H_0</math>): in the closed case, Poincaré duality is the statement that the [[cap product]] with the fundamental class yields an isomorphism <math>H_* (M, R) \to H^{n-*}(M, R)</math>.
 
See also [[Twisted Poincaré duality]]
 
==Applications==
{{Expand section|date=December 2008}}
In the [[Bruhat decomposition]] 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]].
 
==See also==
*[[Longest element of a Coxeter group]]
*[[Poincaré duality]]
 
== External links ==
*[http://www.map.mpim-bonn.mpg.de/Fundamental_class Fundamental class] at the Manifold Atlas.
* The Encyclopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Fundamental_class the fundamental class].
 
{{DEFAULTSORT:Fundamental Class}}
[[Category:Algebraic topology]]

Latest revision as of 09:53, 12 January 2015

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