I am curious about why the definition of $\ker\partial_2$ as 2-cycles (a 2-dimensional hole or void) works so well. It seems like a mystery to me. (I am referring to the simplicial complex case.)
For $n=1$, I can intuitvely understand. For instance, for the loop connecting the vertices $v_0,v_1,v_2$, we have $\partial_1[v_0,v_1]=v_1-v_0$, $\partial_1[v_1,v_2]=v_2-v_1$ and $\partial[v_2,v_0]=v_0-v_2$. So adding them up produces a "telescoping sum" effect, which corresponds to "closing the loop":
$$v_1-v_0+v_2-v_1+v_0-v_2=0$$
For $n=2$, it seems like a mystery to me. Consider the tetrahedron (3 simplex with vertices $v_0,v_1,v_2,v_3$). There are 4 faces:
$$\partial_2[v_0,v_1,v_2]=[v_1,v_2]-[v_0,v_2]+[v_0,v_1]$$ $$\partial_2[v_0,v_1,v_3]=[v_1,v_3]-[v_0,v_3]+[v_0,v_1]$$ $$\partial_2[v_0,v_2,v_3]=[v_2,v_3]-[v_0,v_3]+[v_0,v_2]$$ $$\partial_2[v_1,v_2,v_3]=[v_2,v_3]-[v_1,v_3]+[v_1,v_2]$$
Question: The mystery to me is why is it so nice that:
$$\partial_2[v_0,v_1,v_2]-\partial_2[v_0,v_1,v_3]+\partial_2[v_0,v_2,v_3]-\partial_2[v_1,v_2,v_3]=0$$, which corresponds to the 4 faces enclosing a 2-dimensional hole? What is the mathematics behind it?
I roughly know that this means that the $\partial_2[v_0,v_1,v_2],\dots, \partial_2[v_1,v_2,v_3]$ are "linearly dependent" therefore they have a linear combination that makes zero. However this explanation is not satisfying enough for me.
Further question: This brings me to ask the next question, how about for $n\geq 3$? How are we so sure that $\ker\partial_n$ is a $n$-dimensional hole?
Thanks.