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On page 90 of Calculus on Manifolds, Spivak defines the pull-back $f^*\omega$ of a differential form by a differentiable map by the usual formula. On page 97, he defines a singular $k$-cube as a continuous map $c:[0,1]^k\to\mathbb{R}^n$. Finally, on page 101, he defines the integral of a differential $k$-form against a singular $k$-cube by the formula $\int_c\omega=\int_{[0,1]^k} c^*\omega$.

I don't see, however, where he defines the pull back $c^*\omega$ of a differentiable form by a continuous map (or how he could).

So my questions are: (a) Is this an error (or am I missing something)? (b) Have other people noticed it?

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    $\begingroup$ I would imagine that $c$ is supposed to be $C^\infty$. It might have been an oversight. I don't have a copy to check. $\endgroup$
    – Donu Arapura
    Commented Nov 21, 2011 at 17:45

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It's clearly an elementary oversight that, on the other hand, doesn't matter for the real development of the material. Yes, the chain $c$ ought to be at least $C^1$ for the pullback to be continuous and integrable, and you might as well make it $C^\infty$. You need a smooth approximation lemma that singular (co)homology with smooth chains is the same as singular (co)homology with continuous chains. It is in the spirit of simplicial approximation, which is used to establish isomorphism between combinatorial simplicial (co)homology and singular simplicial (co)homology. You can also use piecewise-smooth chains as a mutual generalization of PL chains and smooth chains.

To see that you have to restrict the chains somehow, you can let $\omega$ be the area form in the plane, and you can use the fact that there is a topological circle in the plane with non-zero area. You can fill in the circle with a 2-chain $c$, and there is no reasonable way to define the pairing $\langle c,\omega \rangle$. This pathology is impossible if both $\omega$ and $c$ are closed, but that is sidestepping the issue, because you can only prove that by setting up enough machinery with smooth restrictions.

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Continuity is not enough. However Lipschitz continuity would do. Lipschitz continuous maps are differentiable almost everywhere and you can define the pullback almost everywhere. For more on this point of view, have a look at a book on geometric measure theory, e.g., Frank Morgan's: Geometric Measure Theory. A Beginner's Guide.

Another beautiful reference is the classic Geometric Integration Theory by H. Whitney (now available in Dover)

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