Using a $\Delta$-complex structure, I've calculated the simplicial homology groups of a Torus.
I used 2 2-simplices, 3 1-simplices and 1 0-simplex under the quotient map. I obtained the following groups:
$$H^{\Delta}_2(T)\simeq{}\mathbb{Z}$$ and $$H^{\Delta}_1(T)\simeq{}\mathbb{Z}^2$$ .
I understand that the $k^{th}$ homology group is the kernel of the boundary map $\Delta^k\to{}\Delta^{k-1}$ 'quotienting out' the image of the boundary map $\Delta^{k+1}\to{}\Delta^{k}$ but im having a hard time understanding what this means geometrically.
For example, in our case of the torus, could I draw on a diagram of the Torus the members of the basis of $H^{\Delta}_1(T)$ for instance. Is there a nice heuristic of what these homology groups are in relation to a given topological space.
EDIT:
or is it not really important, is it more that these groups are invariant when playing around with spaces that are topologically equivalent?