Editing Fundamental class (section)

==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]]