Closed, orientable

When M is a connected orientable closed manifold of dimension n, the top homology group is infinite cyclic: ${\displaystyle H_{n}(M;\mathbf {Z} )\cong \mathbf {Z} }$, and an orientation is a choice of generator, a choice of isomorphism ${\displaystyle \mathbf {Z} \to H_{n}(M;\mathbf {Z} )}$. 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 integration over M; namely for M a smooth manifold, an n-form ω can be paired with the fundamental class as

${\displaystyle \langle \omega ,[M]\rangle =\int _{M}\omega \ ,}$

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, ${\displaystyle H_{n}(M;\mathbf {Z} )\ncong \mathbf {Z} }$, and so one cannot define a fundamental class M living inside the integers. However, every closed manifold is ${\displaystyle \mathbf {Z} _{2}}$-orientable, and ${\displaystyle H_{n}(M;\mathbf {Z} _{2})=\mathbf {Z} _{2}}$ (for M connected). Thus, every closed manifold is ${\displaystyle \mathbf {Z} _{2}}$-oriented (not just orientable: there is no ambiguity in choice of orientation), and has a ${\displaystyle \mathbf {Z} _{2}}$-fundamental class.

This ${\displaystyle \mathbf {Z} _{2}}$-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 ${\displaystyle H_{n}(M,\partial M)\cong \mathbf {Z} }$, and so the notion of the fundamental class can be extended to the manifold with boundary case.