Imagine the grid $\mathbb{Z}^2$ as a simplicial complex such that each square is triangulated into two triangles, that is, the simplices are given by $\{(a, b),(a + 1, b),(a + 1, b + 1)\}$ and $\{(a, b),(a, b + 1),(a + 1, b + 1)\}$ for each $(a,b)\in\mathbb{Z}^2$.
Obviously the homology of this complex is trivial but in dimension zero where it is the integers.
However, I have no idea how to compute the first and second homology group. I really would like to go by definition and make formal that 'every cycle is bounded' and not to use things like homotopy invariance.