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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".
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Definition
Closed, orientable
When M is a connected 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
![\langle\omega, [M]\rangle = \int_M \omega](http://wpcontent.answers.com/math/b/e/e/bee430cbfb3e78d9143ee28a475aee5a.png)
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
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Under Poincaré duality, the fundamental class is dual to the bottom class of a connected manifold (a generator of H0).
Applications
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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
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