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Proceedings of the American Mathematical Society
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 $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0640243-4
PII: S 0002-9939(1982)0640243-4
Article copyright: © Copyright 1982 American Mathematical Society