Editing Fundamental class

{{For|the fundamental class in class field theory|class formation}}
{{More citations needed|date=April 2023}}

In [[mathematics]], the '''fundamental class''' is a [[homology (mathematics)|homology]] class [''M''] associated to a [[Connected space|connected]] [[orientable]] [[closed manifold|compact manifold]] of dimension ''n'', which corresponds to the generator of the homology group <math>H_n(M,\partial M;\mathbf{Z})\cong\mathbf{Z}</math> . The fundamental class can be thought of as the orientation of the top-dimensional [[simplex|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 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 [[differential form|''n''-form]] ω can be paired with the fundamental class as

:<math>\langle\omega, [M]\rangle = \int_M \omega\ ,</math>

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,  <math>H_n(M;\mathbf{Z}) \ncong \mathbf{Z}</math>, and so one cannot define a fundamental class ''M'' living inside the integers. 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 class]].

===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 so the notion of the fundamental class can be extended to the manifold with boundary case.

==Poincaré duality==
{{Main|Poincaré duality}}
{{Expand section|date=December 2008}}
The Poincaré duality theorem relates the homology and cohomology groups of ''n''-dimensional oriented closed manifolds: if ''R'' is a [[commutative ring]] and ''M'' is an ''n''-dimensional ''R''-orientable closed manifold with fundamental class ''[M]'', then for all ''k'', the map 
: <math> H^k(M;R) \to H_{n-k}(M;R) </math>
given by 
: <math> \alpha \mapsto [M] \frown \alpha </math>
is an isomorphism.<ref name=":0" />

Using the notion of fundamental class for manifolds with boundary, we can extend Poincaré duality to that case too (see [[Lefschetz duality]]). In fact, the [[cap product]] with a fundamental class gives a stronger duality result saying that we have isomorphisms <math>H^q(M, A;R) \cong H_{n-q}(M, B;R)</math>, assuming we have that <math>A, B</math> are <math>(n-1)</math>-dimensional manifolds with <math>\partial A=\partial B= A\cap B</math> and <math>\partial M=A\cup B</math>.<ref name=":0">{{Cite book |last=Hatcher |first=Allen |url=https://pi.math.cornell.edu/~hatcher/AT/ATpage.html |title=Algebraic Topology |date=2002 |publisher=[[Cambridge University Press]] |isbn=9780521795401 |edition=1st |location=Cambridge |page= |pages=241–254 |language=English |mr=1867354 |authorlink=Allen Hatcher}}</ref>

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]]

==References==
{{reflist}}

==Sources==
*{{Cite book|first=Allen|last=Hatcher|authorlink=Allen Hatcher|url=https://pi.math.cornell.edu/~hatcher/AT/ATpage.html|title=Algebraic Topology|date=2002|publisher=[[Cambridge University Press]]|isbn=9780521795401|edition= 1st|location=Cambridge|language=English|mr=1867354}}

== 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].

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[[Category:Algebraic topology]]