Editing Fundamental class (section)

==Definition==
===Closed, orientable===
When ''M'' is a [[connected space|connected]] [[orientable]] [[closed manifold]] of dimension ''n'', the top homology group is [[infinite cyclic]]: <math>H_n(M;\mathbf{Z}) \cong \mathbf{Z}</math>, and an orientation is a choice of generator, a choice of isomorphism <math>\mathbf{Z} \to H_n(M;\mathbf{Z})</math>. 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 [[differential form|''n''-form]] ω can be paired with the fundamental class as

:<math>\langle\omega, [M]\rangle = \int_M \omega\ ,</math>

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,  <math>H_n(M;\mathbf{Z}) \ncong \mathbf{Z}</math>, and so one cannot define a fundamental class ''M'' living inside the integers. However, every closed manifold is <math>\mathbf{Z}_2</math>-orientable, and 
<math>H_n(M;\mathbf{Z}_2)=\mathbf{Z}_2</math> (for ''M'' connected). Thus, every closed manifold is <math>\mathbf{Z}_2</math>-oriented (not just orient''able'': there is no ambiguity in choice of orientation), and has a <math>\mathbf{Z}_2</math>-fundamental class.

This <math>\mathbf{Z}_2</math>-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 <math>H_n(M,\partial M)\cong \mathbf{Z}</math>, and so the notion of the fundamental class can be extended to the manifold with boundary case.