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Consider this picture (taken from Hatcher pg 106). Elements of $\Delta_n(X)$, called $n$-chains, can be written as finite formal sums $\sum_\alpha n_\alpha e^n_\alpha$ with coefficients $n_\alpha\in\mathbb{Z}$, and $e^n_\alpha$ are open $n$-simplices.

I can guess that $2e$ means the edge going around anticlockwise twice (is that correct?)

However, I don't know what $2v$ would represent? Or for that matter $-v$?

Thanks for enlightening!

yoyostein
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    It doesn't represent anything more than the formal collection of symbols. It might help to think of $v$ in this case as being a kind of basis vector of a vector space (this is only an analogy as obviously $\mathbb{Z}$ isn't a field, so we only get a $\mathbb{Z}$-module). Then $2v$ just means you have a vector of length $2$ in the `direction' of $v$. – Dan Rust Nov 13 '15 at 15:32

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Formal sums are just that: they are form, not content. You shouldn't think of $2[v]$ as anything more than the number $2$ written next to the symbol $[v]$.

Now, as you realized, there's a special way you can think about $2[e]$ for the case that $[e]$ is an edge (which points from one vertex to itself); loosely, as a path that traces this edge twice. This is a map $\pi_1(X, [e]) \to H_1(X)$ called the Hurewicz map, and indeed "trace the edge twice" maps to $2[e]$ in this case, so the geometric way you've thought about it isn't a complete coincidence.

hunter
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It's a formal sum.

So $2v$ means precisely the coefficient $2$ multiplied by the formal symbol $v$. You add formal sums in much the same way as you add polynomials, termwise. The fact that this is useful might be surprising, but formal sums and formal power series come up quite often as a tool for organizing or structuring information.

davidlowryduda
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