Fundamental class

In mathematics, the fundamental class is a homology class ["M"] associated to a connected orientable compact manifold of dimension "n", which corresponds to the generator of the homology group formula_1 . The fundamental class can be thought of as the orientation of the top-dimensional simplices of a suitable triangulation of the manifold.
Definition.
Closed, orientable.
When "M" is a connected orientable closed manifold of dimension "n", the top homology group is infinite cyclic: formula_2, and an orientation is a choice of generator, a choice of isomorphism formula_3. 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 "n"-form ω can be paired with the fundamental class as
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, formula_5, and so one cannot define a fundamental class "M" living inside the integers. However, every closed manifold is formula_6-orientable, and 
formula_7 (for "M" connected). Thus every closed manifold is formula_6-oriented (not just orient"able": there is no ambiguity in choice of orientation), and has a formula_6-fundamental class.
This formula_6-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 formula_11, and the notion of the fundamental class is extended to the relative case.
Poincaré duality.
For any abelian group formula_12 and non negative integer formula_13 one can obtain an isomorphism 
using the cap product of the fundamental class and the formula_15 -cohomology group . This isomorphism gives Poincaré duality:
Poincaré duality is extended to the relative case .
See also Twisted Poincaré duality
Applications.
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.