# Fundamental class

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In mathematics, the **fundamental class** is a homology class [*M*] associated to an oriented manifold *M*, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold". Intuitively, the fundamental class can be thought of as the sum 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: , 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 a fundamental class for each connected component (corresponding to an orientation for each component).

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 *n*-form ω can be paired with the fundamental class as

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

### Non-orientable

If *M* is not orientable, one cannot define a fundamental class, or more precisely, one cannot define a fundamental class *over * (or over ), as , and indeed, one cannot integrate differential *n*-forms over non-orientable manifolds.

However, every closed manifold is -orientable, and
(for *M* connected). Thus every closed manifold is -oriented (not just orient*able*: there is no ambiguity in choice of orientation), and has a -fundamental class.

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

### With boundary

If *M* is a compact orientable manifold with boundary, then the top relative homology group is again infinite cyclic , and as with closed manifolds, a choice of isomorphism is a fundamental class.

## Poincaré duality

{{#invoke:main|main}} Template:Expand section Under Poincaré duality, the fundamental class is dual to the bottom class of a connected manifold (a generator of ): in the closed case, Poincaré duality is the statement that the cap product with the fundamental class yields an isomorphism .

See also Twisted Poincaré duality

## Applications

Template:Expand section 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

## External links

- Fundamental class at the Manifold Atlas.
- The Encyclopedia of Mathematics article on the fundamental class.