# Boundary of a 2-chain

If $c$ is a singular 1-cube in $\mathbb{R}^2 \backslash \left\{ (0,0)\right\}$ with $c(0)=c(1)$, show that there is an integer $n$ such that: $c-c_{(1,n)} = \partial (c^2)$ for some $2$-chain $c^2$.

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What is "c(subscript:1,n)"? – Robin Chapman Oct 12 '10 at 6:38
The subscript for c is (1,n). I don't know how to use the special characters – JimJones Oct 13 '10 at 3:30
I've been doing some editing to your question: hope I've written what you meant. (You can rollback, if I made any mistake.) But I still don't understand what $c_{(1,n)}$ stands for. As for the tag: I don't see any smooth map there, are you sure that "topology" wouldn't be more correct? – a.r. Oct 13 '10 at 4:42
If you want to learn LaTeX quickly (those "special characters"), click on the words beside "edited" in your question and then on "view source". – a.r. Oct 13 '10 at 4:46
I think what the other commenters mean is that you should try to explain your terminology (like $c_{(1,n)}$) so that others can interpret the question properly. Perhaps provide a motivation for why you're interested in the problem, or a source for where you found it (say, Spivak's "Calculus on Manifolds" Problem 4-24). – Jesse Madnick Oct 13 '10 at 5:58

Ok, so, as Jesse says, this is problem 4-24 from Little Spivak. Maybe you should look at problem 4-23, where he explains what $c_{(1,n)}$ is: a parametrization of the circumference of radius $1$, centered at the origin, that goes round the origin $n$ times

$$c_{(1,n)} (t) = (\cos 2\pi nt, \sin 2\pi nt) \ .$$

So the problem says that, given any closed parametrized curve $c(t)$, there is an integer $n$ such that this $c$ together with $c_{(1,n)}$ are the boundary of some $2$-chain $c^2$.

If I'm not wrong, this $n$ is the winding number of $c$; that is, the number of times $c$ goes round the origin. So, if $c$ goes round the origin one time, you need the circumference $c_{(1,1)}$ that goes round the origin once too; if $c$ goes round twice, you need $c_{(1,2)}$...

Maybe you should try first problem 4-23, where Spivak asks a similar question, but where $c$ is also a circumference (of different radius). There you see too that you need both circumferences to go round the origin the same number of times in order they are the boundary of some $2$-chain. For instance, if $n=1$ (just once round the origin), these circumferences are the boundary of a regular anulus.

Anyway, you may try to take a look at Little Spivak solutions. They are quite right. For instance, in the solution to 4-24, Ken Kubota (the author of these solutions), begins computing $L = \int_0^1 c^*(d\theta)$, which is the winding number of $c$ multiplied by $2\pi$.

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