>>11623872> i need to understand what homology isIt's really not that hard and homology is a general notion that goes beyond just topology.
Assume is some function (homomorphism) d: G->G where G is an abelian group, and assume d^2=0. That tells you that the elements in the image of d are in the kernel of d. But are elements in the kernel in the image of d? Turns out that's not necessarily true, and this naturally leads to the notion of homology, which measures how much this fails.
The homology H is defined to be ker(d)/im(d). This makes sense since im(d) is a subgroup of ker(d), since d^2 =0.
For example, you can take G to be the integers mod 4: Z/4Z and d be d(x)=2x. then d^2(x)=4x=0 for all x, thus we can look at the homology of d.
The kernel of d is just {0,2}, the image of d is also {0,2} so the homology is trvial: H={0}.
Now take G=Z/8Z and d(x)=4x. ker(d)={0,2,4,6} but im(d)={0,4}. We see they are not equal: it's not true that for all numbers n, if 8|4n then n=4k for some k. The homology measures precisely, in a sense, by how much this statement fails. In this case, H=Z/2Z.
