Fundamental class

Closed, orientable

When M is a connected orientable closed manifold of dimension n, the top homology group is infinite cyclic: H n ( M , Z ) ≅ Z , and an orientation is a choice of generator, a choice of isomorphism Z → H n ( M , Z ) . 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 a 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, H n ( M , Z ) ≆ Z , and so one cannot define a fundamental class M living inside the integers. However, every closed manifold is Z 2 -orientable, and H n ( M ; Z 2 ) = Z 2 (for M connected). Thus every closed manifold is Z 2 -oriented (not just orientable: there is no ambiguity in choice of orientation), and has a Z 2 -fundamental class.

This Z 2 -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 H n ( M , ∂ M ) ≅ Z , and the notion of the fundamental class is extended to the relative case.

Poincaré duality

For any abelian group G and non negative integer q ≥ 0 one can obtain an isomorphism

using the cap product of the fundamental class and the q -homology 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.