# Fundamental class

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

### 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 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, , 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 orient*able*: 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

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

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

- .

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

## External links

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