Classification of $3$-manifolds with certain spines

Abstract:

Given the group presentation $\varphi = \left \langle {a,b\backslash {a^m}{b^n},{a^p}{b^q}} \right \rangle$ with $m,n,p,q \ne 0$, we construct the corresponding $2$-complex ${K_\varphi }$. We prove the following theorems. THEOREM 7. ${K_\varphi }$ is a spine of a closed orientable $3$-manifold if and only if (i) $|m| = |p| = 1$ or $|n| = |q| = 1$, or (ii) $(m,p) = (n,q) = 1$. THEOREM 10. If $M$ is a closed orientable $3$-manifold having ${K_\varphi }$ as a spine and $\lambda = |mq - np|$ then $M$ is a lens space ${L_{\lambda ,k}}$ where $(\lambda ,k) = 1$ except when $\lambda = 0$ in which case $M = {S^2} \times {S^1}$.