I'm struggling a bit to understand the concept of a chain in Geometry/Topology, as a linear combination of simplexes, and even more to understand it geometrically (if it possible). So, let's start simply.
Let $\Delta = [a,\,b,\,c]$ be the triangle generated by the points $a,\,b,\,c$ in our space, such that $\partial \Delta$ is the chain $[a,\,b] + [b,\,c] – [a,\,c]$.
If we interpret the boundary $\partial \Delta$ of this triangle as "paths", with a orientation, it's easy to see that $\partial \Delta$ is the polygonal path connecting $a$ to $c$, with $b$ in between. Ok, but what about the chain $[a,\,b] + 2\,[b,\,c] – 5\,[a,\,c]$, how could we interpret it geometrically?
Now, let's go to $\mathbb{R}^n$; considering continuous paths of the form $c_i \colon [0,\,1] \longrightarrow \mathbb{R}^n$, is there a intuitive way of understanding chains of the form $\Gamma = \sum_i n_i \, c_i$, with $n_i \in \mathbb{Z}$?
In my point of view, the integral coeficients of $\Gamma$ implies "how many times" we travel the path $c_i$, i.e., if $\Gamma = 3\,c_1 – c_2 + 2\,c_3$, we travel $3$ times along $c_1$, then one time along $c_1$ (in the opposite way), and then $2$ times along $c_3$.
My goal is to understand properly the Generalized Stoke's Theorem, as well the homologic version of Cauchy's Theorem, and both requires the language of chains. Thanks in advance!
Best Answer
This is not something you should try to overthink. Nonetheless there's a couple of simple straightforward ways that I like to think of the 1-chain $[a,b] + 2 [b,c] - 5 [a,c]$.
One is almost purely formal. First, attached to the oriented segment $[a,b]$ you observe a ghostly floating coefficient $1$. Also, attached to the oriented segment $[b,c]$ you observe a ghostly floating coefficient $2$. And, attached to the oriented segment $[a,c]$, you observe a ghostly floating coefficient $-5$.
Another, less imaginatively spectral way might be like this. When you look closely at the oriented segment $[b,c]$ you actually observe 2 copies of it. When you look closely at the oriented segment $[a,c]$ you actually observe $-5$ copies of it, which leads to the issue of what a "negative" copy means, but perhaps you can think of it as 5 anti-copies, where an "anti"copy and an ordinary copy annihilate each other leaving nothing, just as $-5$ and $5$ add up to nothing. Finally, when you look closely at the oriented segment $[a,b]$ you don't see anything out of the ordinary at all, just $1$ ordinary copy.
And your point of view, counting the how many times to travel, is just fine too.
Oh, and here's one more: think of each simplex as a pipe, and the coefficient as the amount of fluid per unit time flowing through that pipe. So $[a,b] + 2 [b,c] - 5 [a,c]$ is 1 fluid unit per time unit flowing along $[a,b]$ from $a$ to $b$, 2 fluid units per time unit flowing along $[b,c]$ from $b$ to $c$, and $5$ fluid units per time unit flowing along $[a,c]$ from $c$ to $a$. What I like about this point of view is that the boundary of the chain makes physical sense: $$\partial \left( [a,b] + 2 [b,c] - 5 [a,c] \right) = 4a -b -3c $$ means that at $a$ you see 4 fluid units gushing outward per unit time, and at $b$ you see 1 fluid unit sucked inward per unit time; and at $c$ you see 3 fluid units sucked inward per unit time.