I'm reviewing for an upcoming algebraic topology exam, and I have a question that I cannot solve fully.
For $(m,n)\in \mathbb{Z}^2\setminus\{(0,0)\}$, let $X_{m,n}$ be a the CW complex obtained from $S^1$ with its standard cell structure by attaching two 2-cells by maps of degree $m$ and $n$, respectively. Give necessary and sufficient condition under which $X_{m,n}$ and $X_{m',n'}$ are homotopy equivalent.
So far, I have computed the singular homology and found that $$H_n(X_{m,n})=\begin{cases}H_0(X_{m,n})\cong \mathbb{Z}\\ H_1(X_{m,n})\cong \mathbb{Z}/\big(\textrm{gcd}(m,n)\mathbb{Z}\big) \\ H_n(X_{m,n})=0 \textrm{ otherwise }\end{cases}$$ So if $\textrm{gcd(m,n)}\neq\textrm{gcd}(m',n')$, then $X_{m,n}\not\simeq X_{m',n'}$. Next, let $X^1_{m,n}$ denote the one skeleton of our CW complex. A theorem from Hatcher says that if $(X_1,A)$ is a CW complex and $f,g:A\rightarrow X_0$ are homotopic, then $$X_0\sqcup_f X_1\simeq X_0\sqcup _g X_1 $$ relative $X_0$. Now Consider $X_m$ (the CW complex obtained from $S^1$ with it standard cell structure by attached a 2-cell by a map of degree $m$), then $\pi_1(X_m)=\mathbb{Z}_m$ and then we can view $\phi_n(\partial D^2)$ (where $\phi_n$ is an attaching map of degree $n$) as a loop in $\pi_1(X_m)$. We have that $[\phi_n(\partial D^2)]=[n\hspace{1mm}\textrm{mod}\hspace{1mm}m]$. So if $m=m'$ and $n=n'\hspace{1mm}\textrm{mod}\hspace{1mm}m$. Then we have a homotopy equivalence between $X_{m,n}$ and $X_{m',n'}$ because $$X^1_{m,n}\sqcup_{\phi_m}D^2\sqcup_{\phi_n}D^2\simeq X^1_{m,n}\sqcup_{\phi_m}D^2\sqcup_{\phi_{n'}} D^2 $$ by the theroem in Hatcher. The result is the same if $n=n'$ and $m=m'\hspace{1mm}\textrm{mod}\hspace{1mm}n$. My guess would be that I can loosen the requirements on $(m,n)$ and $(m',n')$ and construct a space that would be homotopic to both spaces using maps (of degree gcd$(m,n)$ ?) and show it is homotopic to both $X_{m,n}$ and $X_{m',n'}$ using the theorem mentioned before from Hatcher. However, I don't know how to proceed. Am I headed in the right direction or should I approach the problem differently?