Fundamental class

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For the fundamental class in class field theory, see class formation.

In mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M of dimension n, which corresponds to the generator of the homology group . The fundamental class can be thought of as the orientation of the top-dimensional simplices of a suitable triangulation of the manifold.

Definition[edit]

Closed, orientable[edit]

When M is a connected orientable closed manifold of dimension n, the top homology group is infinite cyclic: , and an orientation is a choice of generator, a choice of isomorphism . 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[edit]

If M is not orientable, , and so one cannot define a fundamental class M living inside the integers. However, every closed manifold is -orientable, and (for M connected). Thus every closed manifold is -oriented (not just orientable: there is no ambiguity in choice of orientation), and has a -fundamental class.

This -fundamental class is used in defining Stiefel–Whitney class.

With boundary[edit]

If M is a compact orientable manifold with boundary, then the top relative homology group is again infinite cyclic , and the notion of the fundamental class is extended to the relative case.

Poincaré duality[edit]

Main article: Poincaré duality

For any abelian group and non negative integer one can obtain an isomorphism

.

using the cap product of the fundamental class and the -homology group . This isomorphism gives Poincaré duality:

.

Poincaré duality is extended to the relative case .

See also Twisted Poincaré duality

Applications[edit]

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.

See also[edit]

External links[edit]