We’ve defined how to integrate forms over chains made up of singular cubes, but we still haven’t really defined integration on manifolds. We’ve sort of waved our hands at the idea that integrating over a cube is the same as integrating over its image, but this needs firming up. In particular, we will restrict to oriented manifolds.
To this end, we start by supposing that an -form is supported in the image of an orientation-preserving singular -cube . Then we will define
Indeed, here the image of is some embedded submanifold of that even agrees with its orientation. And since is zero outside of this submanifold it makes sense to say that the integral over the submanifold — over the singular cube — is the same as the integral over the whole manifold.
What if we have two orientation-preserving singular cubes and that both contain the support of ? It only makes sense that they should give the same integral. And, indeed, we find that
where we use to reparameterize our integral. Of course, this function may not be defined on all of , but it’s defined on , where is supported, and that’s enough.