What does this representation of a chain mean?

I've been learning Stokes' Theorem out of Spivak's Calculus on Manifolds. Toward the end of the section on integration on chains, he defines $n$-cubes and $n$-chains. His definition of $n$-cubes is fairly clear, defined as a map $c:[0,1]^n\rightarrow A$, with $A$ being a general topological space but for the purposes of the text taken to be an open subset of $\mathbb{R}^n$.

At this point he begins to define an $n$-chain as a linear combination of $n$-cubes over the integers, with his canonical example being

$$2c_1+3c_2-4c_3$$

What is this entity supposed to represent? If presented this way my natural belief says that it's another map from $I^n\rightarrow A$, except the text makes it quite unambigiously clear that it is not supposed to be interpreted in this way.

Furthermore, in my calculus course an $n$-chain is defined to be a free $\mathbb{Z}$-module generated by the space of $n$-cubes. If that's the case, then again the above representation doesn't make any sense to me, because then we'd like to have a function that takes cubes as arguments, not the above function which seems to take points in $I^n$ as arguments.

What am I missing? What exactly is the above notation representing?

Solution 1:

You have just a simple misunderstanding: an $n$-chain is not the free $\mathbb{Z}$-module generated by the space of $n$-cubes, but instead is an element of the free $\mathbb{Z}$-module generated by the space of $n$-cubes. By definition elements of the free $\mathbb{Z}$-module generated by any set $S$, which I'll denote $\mathbb{Z}S$, is just the set of formal $\mathbb{Z}$-linear combinations of elements of $S$. Thus if $c_1, c_2, c_3 \in S$ then $2 c_1 + 3 c_2 - 4 c_3$ is certainly an element of $\mathbb{Z}S$. So everything works out and both your book and class are talking about the same thing.

Finally, note that $2 c_1 + 3 c_2 - 4 c_3$ can't be interpreted as just a map $I^n \to A$ as you allude, because $A$ has no notion of addition or subtraction of points.