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

Question

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.

Answer 1

$B$ 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, $$H_2(X)\cong H_2(A)\cong\Bbb Z.$$