Fundamental class

In mathematics, the fundamental class is a homology (mathematics) 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 space, 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 differential form ω can be paired with the fundamental class as

:langleomega, [M] angle = int_M omega

to get a real number, which is the integral of ω over M , and depends only on the cohomology class of ω.