Classification of $3$-manifolds with certain spines
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- by Richard S. Stevens
- Trans. Amer. Math. Soc. 205 (1975), 151-166
- DOI: https://doi.org/10.1090/S0002-9947-1975-0358786-0
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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 $M = {S^2} \times {S^1}$.References
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Bibliographic Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 205 (1975), 151-166
- MSC: Primary 57A10
- DOI: https://doi.org/10.1090/S0002-9947-1975-0358786-0
- MathSciNet review: 0358786