Take the unit disk $D^2$ and take the equivalence relation on the boundary ($S^1$) by identifying the following arcs: $(0, 2\pi/3), (2\pi/3, 4\pi/3), (4\pi/3,2\pi)$ (alternatively identifying points $z, z = e^{2\pi i/3}, z = e^{4\pi i/3}$). Call the space $X$. I have put a delta complex structure on it as follows:
where the outside edges are all identified and inner edges not.
From there, I am taking the ith chain groups to be: $C_2(X) = \mathbb{Z}^3, C_1(X) = \mathbb{Z}^4,$ and $C_0(X) = \mathbb{Z}^2$.
Computing boundary maps (which can be represented as matrices) between each set of chains I have the following maps:
$\begin{bmatrix} 1 & 1 & 1\\ -1 & 0 & 1\\ 1 & -1 & 0\\ 0 & 1 & -1\end{bmatrix}:C_2 \rightarrow C_1$ and $\begin{bmatrix} 1 & 1 & 1 & 1\\ 0 & 1 & 1 & 1\end{bmatrix}:C_1 \rightarrow C_0$.
First of all, I'm not even sure the second map is correct (how should I count signs here?). Secondly, once we correct those maps, I know the general idea is to compute the kernel of the maps and take quotients but I'm extremely shaky in that area and would appreciate any help.
edit: I've realized the second map is likely the following:
$\begin{bmatrix} 0 & 1 & 1 & 1\\ 0 & -1 & -1 & -1\end{bmatrix}:C_1 \rightarrow C_0$
As a follow up, I believe I have the kernels of our two maps: The first map has a trivial kernel, while the second is generated by the following: $\left<\begin{bmatrix} 1\\ 0\\ 0\\ 0\end{bmatrix}, \begin{bmatrix} 0\\ 2\\ -1\\ -1\end{bmatrix} \right>$, i.e. it might look like $\mathbb{Z}^2$?