On genus Heegaard diagrams for the -sphere

Author:
Takeshi Kaneto

Journal:
Trans. Amer. Math. Soc. **276** (1983), 583-597

MSC:
Primary 57M40; Secondary 20F05, 57M05

DOI:
https://doi.org/10.1090/S0002-9947-1983-0688963-5

MathSciNet review:
688963

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be any genus Heegaard diagram for the -sphere and be the cyclically reduced presentation associated with . We shall show that contains or as a subword in cyclic sense if holds, and that, using this property, can be transformed to the trivial one . By the recent positive solution of genus Poincaré conjecture, our result implies the purely algebraic, algorithmic solution to the decision problem; whether a given -manifold with a genus Heegaard splitting is simply connected or not, equivalently, is homeomorphic to the -sphere or not.

**[1]**J. S. Birman,*Heegaard splitting, diagrams and sewings for closed orientable*-*manifolds*, Lecture notes for CBMS conference, October 8-12, 1977.**[2]**J. S. Birman and H. M. Hilden,*Heegaard splittings of branched coverings of*, Trans. Amer. Math. Soc.**213**(1975), 315-352. MR**0380765 (52:1662)****[3]**T. Homma,*On presentations of fundamental groups of*-*manifolds of genus*, Res. Rep. Inf. Sci. Ser. A. Math. Sci. Tokyo Inst. of Tech. No. A-58 (1978).**[4]**T. Homma, M. Ochiai and M. Takahashi,*An algorithm for recognizing**in*-*manifolds with Heegaard splittings of genus two*, Osaka J. Math.**17**(1980), 625-648. MR**591141 (82i:57013)****[5]**T. Kaneto,*On presentations of the fundamental group of the*-*sphere associated with Heegaard diagrams*, J. Math. Soc. Japan**33**(1981), 147-158. MR**597485 (82h:57003)****[6]**O. Morikawa,*A counter-example to a conjecture of Whitehead*, Math. Sem. Notes Kobe Univ.**8**(1980), 295-299. MR**601897 (82h:57011)****[7]**M. Takahashi,*An alternative proof of Birman-Hilden-Viro's theorem*, Tsukuba J. Math.**2**(1978), 27-34. MR**531959 (80e:57009)****[8]**O. Ja. Viro and V. L. Kobel'skii,*The Volodin-Kuznetsov-Fomenko conjecture on Heegaard diagrams is false*, Uspehi Mat. Nauk**32**(1977), no. 5(197), 175-176. MR**0467757 (57:7609)****[9]**I. A. Volodin, V. E. Kuznetsov and A. T. Fomenko,*The problem of discriminating algorithmically the standard three-dimensional sphere*, Russian Math. Surveys**29**(1974), 71-172.**[10]**J. H. C. Whitehead,*On certain sets of elements in a free group*, Proc. London Math. Soc.**41**(1936), 48-56.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0688963-5

Keywords:
-sphere,
-manifolds,
Heegaard diagrams of genus ,
fake Heegaard diagrams,
surgeries,
wave,
group presentations,
substitutions,
strongly simply trivial

Article copyright:
© Copyright 1983
American Mathematical Society