Finding the homology group of $H_n (X,A)$ when $A$ is a finite set of points It is one of the problems in Hatcher's book.
I need to find the homology group of $H_n (X,A)$ when $A$ is a finite set of points and $X$ is $S^2$ or $T^2$. 
I figured out that for $n>1$, I could use the long exact sequence and make $H_n (X,A)$ isomorphic to $H_n (X)$. 
However, I am stuck with $H_1 (X,A)$ and $H_0 (X,A)$.
Can anyone give me an idea how I can find these?
 A: This uses Proposition 2.22 in Hatcher(and you must prove $(X,A)$ is a good pair).
Without saying too much, although I guess "you" did, I solved this by finding a homotopy equivalence between the space $S^2/A$ or $T^2/A$ and a CW complex. In the former case, you can see the homotopy between $S^2/A$ and a CW complex given by $k+1$ 0-cells(where $A$ is a collection of $k$ points), $2k$ 1-cells, and 2 2-cells. Graphically we arrange the first $k$ 0-cells into a lovely regular $k$-gon, with an outlier $x$ in the back. We use the first $k$ 1-cells to add edges to our $k$-gon, and the second $k$ connecting the vertices of the $k$-gon to the outlier. Then the two cells are each attached with their boundaries glued to the $k$-gon. By contracting the 1-cells attached to $x$, we see this is homotopic to $S^2/A$, but this has a CW complex structure, so it is easier to compute.
A: Suppose $A=\{x_1,\dots,x_k\}$.  Then $H_0(A)=\mathbb{Z}^k$ and $H_i(A)=0$ for $i>0$.
Whether $X=S^2$ or $T^2$ we have $H_0(X)\cong\mathbb{Z}$, and like Matt N said in his comment in either case $H_0(X,A)\cong\tilde{H}_0(X/A)=0$.
If $X=S^2$ then $H_1(X)=0$ so the l.e.s. has a portion like $$0\rightarrow H_1(X,A)\rightarrow \mathbb{Z}^k\rightarrow \mathbb{Z}\rightarrow 0 $$ and so $H_1(S^2,A)\cong\mathbb{Z}^{k-1}$.
If $X=T^2$ then $H_1(X)=\mathbb{Z}^2$, so we have $$ 0\rightarrow \mathbb{Z}^2\rightarrow H_1(X,A)\stackrel{\partial}{\rightarrow} \mathbb{Z}^k\rightarrow \mathbb{Z}\rightarrow 0$$
Then $\ker\partial\cong\mathbb{Z}^2$ and its image is $\cong\mathbb{Z}^{k-1}$.  I believe this is enough to conclude $H_1(T^2,A)\cong \mathbb{Z}^{k+1}$
