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EspañolFUNDAMENTAL CLASS
(Redirected from Fundamental homology class)
In mathematics, the 'fundamental class' is a homology class [''M''] associated to a manifold ''M''. It is defined (firstly) in cases when ''M'' is a closed manifold of dimension ''n'', and oriented. It is then an element of ''Hn''(''M'','Z'). If ''M'' is connected, that group is infinite cyclic, and it is the generator picked out by the given orientation.
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.
In mathematics, the 'fundamental class' is a homology class [''M''] associated to a manifold ''M''. It is defined (firstly) in cases when ''M'' is a closed manifold of dimension ''n'', and oriented. It is then an element of ''Hn''(''M'','Z'). If ''M'' is connected, that group is infinite cyclic, and it is the generator picked out by the given orientation.
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.
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