Suppose I am given an exact sequence:
$$0\to G\xrightarrow{f} \mathbb{Z} \xrightarrow{g} \mathbb{Z} \xrightarrow{h} H\to 0 $$
where the first $\mathbb{Z}=H_3(A\cup B)$ and the second $\mathbb{Z}=H_2(A\cap B)$ where $A,B$ are solid tori. I compute the homology of $A\cup B$ and $A \cap B$ by noting that $A\cup B=S^3, A\cap B=$ Torus.
I know by exactness that $f$ should be injective, i.e. $ker(f)=0$ and $h$ should be surjective, i.e. $im(h)=H$. But I got stuck when computing $G$ and $H$.
Any help is appreciated :)