Find a relation between $H_1(B)\to H_1(X)$ of the following pair $(X,B)$

Question

Let $X$ be the 2-dimensional torus minus an open disk. Let $B$ be the boundary of $X$. Find the relation between the homology groups $H_1(B)\to H_1(X)$ induced by the inclusion $B\hookrightarrow X$.

Here is my try:

By a sequence of homology groups that can be found in p.117 of Hatcher's book, we know that there exists an exact sequence: $$\ldots \longrightarrow H_2(X,B) \longrightarrow H_1(B)\stackrel{i_\star}{\longrightarrow}H_1(X)\longrightarrow H_1(X,B)\longrightarrow \ldots $$

Also, the pair $(X,B)$ is a good pair and $X/B$ is a torus. This implies that the last sequence is equivalent to the following: $$\ldots \longrightarrow\mathbb Z \longrightarrow H_1(B)\stackrel{i_\star}{\longrightarrow}H_1(X)\longrightarrow \mathbb Z_2\longrightarrow \ldots $$

Now I am stuck. What can I conclude from the map $i_\star$?

Thanks in advance

Answer 1

Consider the usual picture of the torus as a rectangle with edges identified. Question #1: do you know how the generators of $H_1(X)$ might be represented in this picture? Now suppose that the "open disk" is actually a slightly smaller rectangle, so $B$ is the boundary of that rectangle. $B$ is homeomorphic to a circle, so you should be able to compute its homology. Question #2: can you see what $i_{\star}$ does on the elements of $H_1(B)$?