Heegaard diagrams of lens spaces
Author:
R. P. Osborne
Journal:
Proc. Amer. Math. Soc. 84 (1982), 412414
MSC:
Primary 57M05; Secondary 57N10
MathSciNet review:
640243
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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]
Friedhelm
Waldhausen, Some problems on 3manifolds, Algebraic and
geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford,
Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc.,
Providence, R.I., 1978, pp. 313–322. MR 520549
(80g:57013)
 [W2]
Friedhelm
Waldhausen, HeegaardZerlegungen der 3Sphäre, Topology
7 (1968), 195–203 (German). MR 0227992
(37 #3576)
 [B&M]
Joan
S. Birman and José
María Montesinos, On minimal Heegaard splittings,
Michigan Math. J. 27 (1980), no. 1, 47–56. MR 555836
(81b:57007)
 [S]
Richard
S. Stevens, Classification of 3manifolds with
certain spines, Trans. Amer. Math. Soc. 205 (1975), 151–166.
MR
0358786 (50 #11245), http://dx.doi.org/10.1090/S00029947197503587860
 [O&S]
R.
P. Osborne and R.
S. Stevens, Group presentations corresponding to spines of
3manifolds. I, Amer. J. Math. 96 (1974),
454–471. MR 0356058
(50 #8529)
 [W1]
 F. Waldhausen, Some problems on manifold, Proc. Sympos. Pure Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1977, pp. 313322. MR 520549 (80g:57013)
 [W2]
 , HeegaardZerlegungen der Sphäre, Topology 7 (1968), 195203. MR 0227992 (37:3576)
 [B&M]
 J. Birman and J. Montesinos, On minimal Heegaard splittings, Michigan Math. J. 27 (1980), 4757. MR 555836 (81b:57007)
 [S]
 R. S. Stevens, Classification of manifolds with certain spines, Trans. Amer. Math. Soc. 205 (1975), 151166. 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), 454471. MR 0356058 (50:8529)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
57M05,
57N10
Retrieve articles in all journals
with MSC:
57M05,
57N10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198206402434
PII:
S 00029939(1982)06402434
Article copyright:
© Copyright 1982
American Mathematical Society
