I just started reading about the general idea behind homology theory (Hatcher). During his introduction he is computing the Homology groups $H_n(X)$ of a given space $X$ by building the quotient of the generators of $\ker \partial_n$ and $\operatorname{Im} \partial_{n+1}$.
Unfortunately, my algebra background is not too solid and therefore i don't know how exactly the quotient of two given generators is supposed to be computed.
Intuitively, i understood that $$\langle a, b, c\rangle/\langle a,b\rangle = \langle c\rangle $$ or $$(\mathbb{Z} + \mathbb{Z})/\mathbb{Z} = \mathbb{Z}$$
However, that was mostly my intuitive guess and i do not know what the algebraic reasons behind the solutions are. I tried googling but haven't gotten far.
I basically just need a hint or advice what theorem/topic/chapter i need to study in order to learn the algebraic background for this.
Thanks for any help.
Best Answer
In addition to the first isomorphism theorem as mentioned by @Berci, another tool is the classification of finitely generated abelian groups. The kernel of any boundary map is a subgroup of a finitely generated free abelian group, and hence is finitely generated free abelian. Then the homology is a quotient of that, so it's a finitely generated abelian group. So you could read about this classification theorem and also Smith Normal Form. Visit https://www.williamstein.org/papers/ant/html/node9.html, for example.