recently I've jumped into Topological Data Analysis (TDA) and I'm trying to get some insights about what's behind it in terms of math, in particular regarding simplicial Homology. I'll briefly recap something based on my understanding:
Given a Simplicial Complex $X$, the boundary map $\partial_n : \Delta_n(X) \rightarrow \Delta_{n-1}(X)$ is defined on the basis elements by
$$\partial_n([v_0, \dots, v_n]) = \sum_{i=0}^n(-1)^i[v_0, \dots, \hat{v_i}, \dots, v_n]$$
This operator is extracting the boundary of a chain with respect to the given orientation. (The $\hat{}$ notation means removing the $i$-th vertex and hence obtaining a $n-1$ simplex.
$\Delta_n(X)$ can be seen both as a group, or even as a vector space, if we consider the coefficients of the above sum belonging to some field (e.g. $\mathbb{Z}_2$).
Then, for each dimension one can consider
$$\dots \longrightarrow \Delta_{n+1}(X) \rightarrow \Delta_n(X) \rightarrow \dots \rightarrow \Delta_1(X) \rightarrow \Delta_0(X) \rightarrow 0$$
by iteratively applying the boundary operator $\partial_i : \Delta_i(X) \rightarrow \Delta_{i-1}(X)$.
For each $n$, let
\begin{align} &Ker \, \, \partial_n = \{c \in \Delta_n(X) \, : \, \partial_n \, c = 0\}\\ &Im\,\,\partial_{n+1} = \{c \in \Delta_n(X) \, : \, c = \partial_{n+1}c', \, \, c' \in \Delta_{n+1}(X)\} \end{align}
From the above chain it is clear that $Im\,\partial_{n+1} \subset Ker \partial_n$, and finally ,
The $n$- dimensional homology of $X$ is defined as $H_n = Ker \,\, \partial_n / Im \partial_{n+1}$
What is supposed to be interesting now is looking at the dimension of this quotient vector space by varying $n$ and obtaining what are known as Betti numbers which should give some geometrical information. In particular,
$H_0$ gives the number of connected components
$H_1$ gives the numbers of holes
$H_2$ gives number of 'cavities'..
What I don't understand is exactly the connection between the theory we have constructed until now and these final geometric consideration depending on the dimension/rank of $H_n$.
For example, $H_0 = Ker \partial_0 / Im \, \partial_1 $ where $\partial_{0}$ is taking all $0$-th simplices (or points) and bringing them to $0$, while $\partial_{1}$ is taking all $1$-simplices and bringing them to points.. where is here, for example, the link to the connected components?
EDIT:
Let's consider an element of the 1-chain $\Delta_1(X)$, say $c$, its boundary operator $\partial_1 \, c = [v_1] -[v_0]$. Then, the image of $\partial_1$ are clearly all the points $y-x$ while $Ker \, \partial_0$ are all $0$-simplices $\sigma_i$, belonging to $0$-chain $\Delta_0(X)$ such that $\sum_{i}\alpha_i\sigma_i=0$.
So the elements of quotient should be all objects such that $\sum_i \alpha_i\sigma_i =0$ and this means taking $y=x$ on the subspace we are quotienting to, so somehow 'closing'.
These are just some of my messy thoughts, am I on the right path? What about higher dimensions?
Many thanks in advance for your help,
James