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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Heegaard diagrams of lens spaces

Author: R. P. Osborne
Journal: Proc. Amer. Math. Soc. 84 (1982), 412-414
MSC: Primary 57M05; Secondary 57N10
MathSciNet review: 640243
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Abstract: Let $ (M,F;\upsilon ,w)$ be a Heegaard diagram of $ M$. The complexity of this diagram is the number of points in $ \upsilon \cap m$. This is also the length of the relators in a group presentation naturally corresponding to this diagram. We give an example to show that a Heegaard diagram of minimal complexity need not have a cancelling pair of meridian disks. In terms of the presentation, this says that a minimal length presentation need not have a defining relator for one of the generators. This provides a counterexample to a conjecture of Waldhausen. Our example depends on the rather trivial observation that the shortest possible $ 2$-generator presentation of the cyclic group of order $ 173$ is $ \left\langle {a,b\vert{a^{13}}{b^2},{a^{ - 2}}{b^{13}}} \right\rangle $.

References [Enhancements On Off] (What's this?)

  • [W1] F. Waldhausen, Some problems on $ 3$-manifold, Proc. Sympos. Pure Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1977, pp. 313-322. MR 520549 (80g:57013)
  • [W2] -, Heegaard-Zerlegungen der $ 3$-Sphäre, Topology 7 (1968), 195-203. MR 0227992 (37:3576)
  • [B&M] J. Birman and J. Montesinos, On minimal Heegaard splittings, Michigan Math. J. 27 (1980), 47-57. MR 555836 (81b:57007)
  • [S] R. S. Stevens, Classification of $ 3$-manifolds with certain spines, Trans. Amer. Math. Soc. 205 (1975), 151-166. MR 0358786 (50:11245)
  • [O&S] R. P. Osborne and R. S. Stevens, Group presentations corresponding to spines of $ 3$-manifolds. I, Amer. J. Math. 96 (1974), 454-471. MR 0356058 (50:8529)

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Article copyright: © Copyright 1982 American Mathematical Society

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