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In "Real and complex analysis" a chain is defined as "a sort of formal sum of paths". Given a family $\lbrace\gamma_1,\ldots,\gamma_n\rbrace$ of paths, the chain $\Gamma$ corresponding to these paths is "a sort of formal sum $\Gamma=\gamma_1\dot+\cdots\dot+\,\gamma_n$" and the line integral of this is defined as follows: $$ \int_\Gamma{f\,\mathrm{d}z}:=\int_{\gamma_1}{f\,\mathrm{d}z}\,+\cdots+\int_{\gamma_n}{f\,\mathrm{d}z} $$ I don't like such pseudo definitions since they tell me what I can do with an object, but not what it actually is. So what is a chain really? Thanks

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    $\begingroup$ It's an element of the free abelian group whose basis is the set of all paths. $\endgroup$ – PersonX Apr 11 '13 at 14:06
  • $\begingroup$ @Chris: I am not convinced that it should be the free abelian group as opposed to just the free monoid over the set of all paths. :-) $\endgroup$ – Willie Wong Apr 11 '13 at 14:10
  • $\begingroup$ @WillieWong Don't we want to take orientation into account, which is where the inverse structure comes in? $\endgroup$ – Alex Youcis Apr 11 '13 at 14:20
  • $\begingroup$ @AlexYoucis: yes and no, depending on how we look at it; in any rate, I don't think it's good to see it as a free abelian group. I'm writing an answer to reflect on that right now. $\endgroup$ – tomasz Apr 11 '13 at 14:22
  • $\begingroup$ @Alex: I was being partly facetious. But the phrase "a sort of formal sum..." does not necessarily suggest an orientation. For example, a very similar definition can be made about the line integral in the real case, where the integral is not direction dependent. $\endgroup$ – Willie Wong Apr 11 '13 at 14:25
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This is more in the abstract-algebra territory rather than complex-analysis (especially not the complex part; the definition of path integral does not really depend on complex structure), but I see two (or three) ways of properly formalizing that.

One way is to follow Willie Wong's suggestion, and take for chains the free monoid with a basis of (sufficiently well-behaved, for instance $C^1$) paths, that is, a $\Gamma$ is a chain if it is a finite sequence of paths, and we add chains by appending one at the end of another; for the remainder of this post, call this monoid $Ch_1$

Another, closely related, is to take it to be instead the free commutative monoid defined by the nice paths -- this is the same as above, except we forget the ordering of the sequence. This is the most literal reading of the statement "formal sum" -- a formal sum of things is, to me anyway, the commutative monoid generated by them. If we call it $Ch_2$ we can see that this is just $Ch_1$ divided by the relation $\Gamma_1\Gamma_2=\Gamma_2\Gamma_1$.


A third, quite a bit more complicated way to interpret it is to take into account what we actually do with the chains: we integrate along them. Furthermore, if we restrict ourselves to nice functions (for instance, globally continuous), we can see that for two very different chains $\Gamma,\Gamma'$ we have $\int_\Gamma$ being the same thing as $\int_{\Gamma'}$ (as linear functionals on the space of continuous functions), for example if $\Gamma'$ is just a subdivision of $\Gamma$.

One way to formalize it is to just equate a chain $\Gamma$ with the integration operator $\int_\Gamma$, effectively seeing the chains just a subgroup $Ch_3$ of the additive group of the dual of the space of continuous functions, $C({\bf R}^n)^*$ (or $C({\bf C}^n)^*$). This means that in particular, we see that the chain consisting of a path $\gamma$ and the path $-\gamma$ obtained by reversing the orientation on $\gamma$ is equated to an empty path – that is because for a continuous function $f$ we have that $\int_\gamma f+\int_{-\gamma} f=0$.

While the reference to the dual of continuous functions may seem rather technical, this is, I believe, the intuitive way to see chains. I think that's where the convention to write $-\gamma$ for an inverted path comes from in the first place.

There is a minor gripe for it: if we integrate functions that are not globally continuous, for example ones that have singularities, such as $z^{-1}$ on ${\bf C}$, we will have pairs of chains that are equivalent, but such that we can integrate along one, but not the other, for example if we take $\gamma$ to be the interval from $-1$ to $1$ on the complex plane, then the equation $\int_\gamma z^{-1}+\int_{-\gamma} z^{-1}=0$ doesn't make sense. This wasn't a problem at all in case of $Ch_1,Ch_2$, since in those ones we never really cancelled any terms.

But it's not that much of a problem in this case, as equivalent chains for which the integral is well defined will integrate a continuous function consistently, even if it is not globally defined.

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