Back to Chain Groups¶
Sigh. Ok, that was a lot of stuff we had to get through, but now we’re back to the real problem we care about: figuring out the homology groups of a simplicial complex. As you may recall, we had left off discussing chain groups of a simplicial complex. I don’t want to have to repeat everything, so just scroll up and re-read that part if you forget. I’ll wait…
Let $\mathcal S = \text{{{a}, {b}, {c}, {d}, {a, b}, {b, c}, {c, a}, {c, d}, {d, b}, {a, b, c}}}$ be an oriented abstract simplicial complex (depicted below) constructed from some point cloud (e.g. data set). The n-chain, denoted $C_n(S)$ is the subset of $S$ of $n$-dimensional simplicies. For example, $C_1(S) = \text{ {{a, b}, {b, c}, {c, a}, {c, d}, {d, b}}}$ and $C_2(S) = \text{{a, b, c}}$.
Now, an n-chain can become a chain group if we give it a binary operation called addition that satisfies the group axioms. With this structure, we can add together $n$-simplicies in $C_n(S)$. More precisely, an $n$-chain group is the sum of $n$-chains with coefficients from a group, ring or field $F$. I’m going to use the same $C_n$ notation for a chain group as I did for an n-chain. where $\sigma_i$ refers to the $i$-th simplex in the n-chain $C_n$, $a_i$ is the corresponding coefficient from a field, ring or group, and $S$ is the original simplicial complex.
Technically, any field/group/ring could be used to provide the coefficients for the chain group, however, for our purposes, the easiest group to work with is the cyclic group $\mathbb Z_2$, i.e. the integers modulo 2. $\mathbb Z_2$ only contains ${0,1}$ such that $1+1=0$ and is a field because we can define an addition and multiplication operation that meet the axioms of a field. This is useful because we really just want to be able to either say a simplex exists in our n-chain (i.e. it has coefficient of $1$) or it does not (coefficient of $0$) and if we have a duplicate simplex, when we add them together they will cancel out. It turns out this is exactly the property we want. You might object that $\mathbb Z_2$ is not a group because it doesn’t have an inverse, e.g. $-1$, but in fact it does, the inverse of $a$, for example, is $a$. Wait what? Yes, $a = -a$ under $\mathbb Z_2$ because $a + a = 0$. That’s all that’s required for an inverse to exist, you just need some element in your group such that $a+b=0; \forall a,b \in G$ ($G$ being a group).
If we use $\mathbb Z_2$ as our coefficient group, then we can essentially ignore simplex orientation. That makes it a bit more convenient. But for completeness sake, I wanted to incorporate orientations because I’ve most often seen people use the full set of integers $\mathbb Z$ as coefficients in academic papers and commercially. If we use a field with negative numbers like $\mathbb Z$ then our simplices need to be oriented, such that $[a,b] \neq [b,a]$. This is because, if we use $\mathbb Z$, then $[a,b] = -[b,a]$, hence $[a,b] + [b,a] = 0$.