Heegaard diagrams of lens spaces

Author:
R. P. Osborne

Journal:
Proc. Amer. Math. Soc. **84** (1982), 412-414

MSC:
Primary 57M05; Secondary 57N10

DOI:
https://doi.org/10.1090/S0002-9939-1982-0640243-4

MathSciNet review:
640243

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Abstract: Let be a Heegaard diagram of . The complexity of this diagram is the number of points in . 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 -generator presentation of the cyclic group of order is .

**[W1]**F. Waldhausen,*Some problems on**-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**-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**-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**-manifolds*. I, Amer. J. Math.**96**(1974), 454-471. MR**0356058 (50:8529)**

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DOI:
https://doi.org/10.1090/S0002-9939-1982-0640243-4

Article copyright:
© Copyright 1982
American Mathematical Society