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

MathSciNet review:
688963

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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.

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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