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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Classification of $ 3$-manifolds with certain spines


Author: Richard S. Stevens
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
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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) $ \vert m\vert = \vert p\vert = 1$ or $ \vert n\vert = \vert q\vert = 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 = \vert mq - np\vert$ 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 [Enhancements On Off] (What's this?)

  • [1] R. H. Bing, Mapping a 3-sphere onto a homotopy 3-sphere, Topology Seminar (Wisconsin, 1965) Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N.J., 1966, pp. 89–99. MR 0219071
  • [2] Edwin E. Moise, Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96–114. MR 0048805, https://doi.org/10.2307/1969769
  • [3] L. Neuwirth, An algorithm for the construction of 3-manifolds from 2-complexes, Proc. Cambridge Philos. Soc. 64 (1968), 603–613. MR 0226642
  • [4] -, Some algebra for $ 3$-manifolds, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 179-184. MR 43 #2716.
  • [5] K. Reidemeister, Homotopieringe und Linsenraume, Abh. Math. Sem. Univ. Hamburg 11 (1935), 102-109.
  • [6] H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, Leipzig, 1934; reprint, Chelsea, New York, 1947.
  • [7] Heiner Zieschang, Über einfache Kurvensysteme auf einer Vollbrezel vom Geschlecht 2, Abh. Math. Sem. Univ. Hamburg 26 (1963/1964), 237–247 (German). MR 0161316, https://doi.org/10.1007/BF02992790
  • [8] H. Cišang, Simple path systems on full pretzels, Mat. Sb. (N.S.) 66 (108) (1965), 230–239 (Russian). MR 0193633

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0358786-0
Keywords: Cell-complex, spine, regular neighborhood, collapsing, $ 2$-complex of group presentation, lens space
Article copyright: © Copyright 1975 American Mathematical Society

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