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".

[edit] 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.

[edit] 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 (if M is connected), 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 orientable: there is no ambiguity in choice of orientation), and has a -fundamental class.

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

[edit] With boundary

If M is an 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.