I want to compute the relative homology groups $H_q(X,A)$, where $X=S^1\times[0,1]$ is a cylinder and $A=S^1\times\{0,1\}$ is the space containing the two bounding circles of $X$.
I showed that $H_q(X)=H_q(A)=0$ for $q>1$ and $H_1(X)=H_0(X)=\mathbb Z$ and $H_1(A)=H_0(A)=\mathbb Z^2$. This gave me $H_q(X,A)=0$ for $q>2$ and an elementary theorem states $H_0(X,A)=0$.
However, the cases $q=1$ and $q=2$ trouble me. The long exact homology sequence is $$0\to H_2(X,A)\to\mathbb Z^2\to\mathbb Z\to H_1(X,A)\to\mathbb Z^2\to\mathbb Z\to 0.$$ But from this alone, I cannot uniquely determine the homology groups (if I am correct, they must have the same rank and the rank is $\leq 2$). I could look at the groups themselves (rather than only relying on the exactness of the sequence above), but don't really know how to do that.