# homology groups of a torus with a disk glued to it in a certain way

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.

$$B$$ is contractible, so you have $$H_1(B)=H_2(B)=0$$. You also have $$H_2(A\cap B)=0$$ since $$A\cap B$$ is homotopy equivalent to a circle. So you have an exact sequence $$H_1(A\cap B)\stackrel{h}\to H_1(A)\to H_1(X)\to H_0(A\cap B)\stackrel{k}\to H_0(A)\oplus H_0(B).$$ The map $$k$$ is injective, and effectively $$h$$ is the map $$\Bbb Z\to\Bbb Z^2$$ taking $$1$$ to $$(4,-2)$$. Therefore $$H_1(X)\cong\frac{\Bbb Z^2}{\{(4a,-2a):a\in\Bbb Z\}}\cong \Bbb Z\oplus\frac{\Bbb Z}{2\Bbb Z}.$$
Another stretch of the exact sequence is $$0\to H_2(A)\to H_2(X)\to H_1(A\cap B)\stackrel{h}\to H_1(A)$$ and as $$H$$ is injective, $$H_2(X)\cong H_2(A)\cong\Bbb Z.$$