Take $M_p$ the mapping cylinder (MC) of the $p=3$-fold cover of $S^1$, $M_q$ the MC of the $q=2$-fold cover of $S^1$, where for both, the identification of the MC is done on the side $\{0\}$ of $[0,1]$. Then $M=M_p\cup_i M_q$ where the isomorphism $i$ identify $S^1\times \{1\}$ in $M_p$ and $M_q$.

Does $M$ have a name ? Is there some literature on this object, studying homotopy and homology groups ? Same question if we change $p$ and $q$ by arbitrary integers.