I am trying to study for my qualifying exams and I was trying to solve this problem. So the idea is to form a topological space X by attaching a disk $D^2$ along its boundary to the torus $T^2$ so that the boundary is attached to a loop representing the homology class $4[\alpha]-2[\beta]$ in $T^2$. And we need to calculate the homology groups of X.
My approach: So I used Mayer-vietoris sequence by taking A and B respectively to be neighborhoods of $D^2$ and $T^2$ respectively. Then $ A \cap B$ is the circle $S^1$. I have used the reduced mayer-vietoris sequence. Here's my problem, we need the map $h: H_1( A \cap B) \mapsto H_1(A)+H_1(B) $. I know that $h([\gamma])=0 + 4[\alpha]-2[\beta]$. I think $h$ is injective.
Alternatively, does anyone have an idea how to solve this using cellular homology? I would appreciate both methods so I can compare them.