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(Graph homology, from this Wikipedia page: https://en.wikipedia.org/wiki/Graph_homology)

I'm interested in the full Cayley table and geometric representation of the free abelian group $\mathfrak C_1$ generated on a graph $$\mathfrak G = \bigg( x \xrightarrow{\quad a \quad} y \xrightarrow{\quad b \quad} z \xrightarrow{\quad c, d \quad} x \bigg)$$

Which can be drawn as

enter image description here

Wikipedia adds a group structure to the edges such that we have $\mathfrak C_1 = \big( \{ a, b, c, d \}, + \big)$. Let blue (positive) be the clockwise direction for the edges, and red (negative) be the direction corresponding to counter-clockwise. Of course, we don't need this orientation for the graph, but it corresponds to what's in the figure. An obvious expression would be $a + b + c $, which is just going in a loop around the graph:

enter image description here

More challenging geometrically would be $a + c$, as these two elements are not "connected" as end-to end:

enter image description here

You could try to "abuse" commutativity, and have $a + c = c + a = $ "The path from z to y", but it already feels like shaky ground.

Now, have $a - c$

enter image description here

These paths "diverge" and point to two different places, yet by the group axioms there must be some element within the group that they point two. I'd like two things if possible, which can either be provided by a user or can have a link to their information shared:

(i)

What group is this? Is there a full Cayley table available online that I can look at?

(ii)

What is the geometric interpretation of each element in the Cayley table? What underlying geometric property gives rise to the fact that this is an abelian group? I'd prefer no answers to this question rather than answers that try to claim it's only a formal operation, as those answers tend to be wrong.


Thanks in advance for any helpful links.

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  • $\begingroup$ Do you know what a free abelian group is? $\endgroup$
    – Randall
    Feb 10 at 2:20
  • $\begingroup$ Curious: why are answers that claim it's only a formal operation "wrong"? That is quite literally the way one defines a free object from a generating set from scratch, and is pretty foundational to gobs of modern mathematics, even though it's "only" a formal operation. $\endgroup$
    – Randall
    Feb 10 at 2:24
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    $\begingroup$ I'll let someone else--better at explaining--jump in. $\endgroup$
    – Randall
    Feb 10 at 2:48
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    $\begingroup$ @Amateur_Algebraist Thanks for noticing the typo. Corrected. $\endgroup$
    – Nate
    Feb 10 at 9:07
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    $\begingroup$ There is a lot of geometric motivation for (simplicial, singular) homology. I admit it's a bit lost from modern sources. I don't go into all of it but maybe this helps $\endgroup$
    – FShrike
    Feb 10 at 11:27

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