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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On genus $ 2$ Heegaard diagrams for the $ 3$-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 $ D$ be any genus $ 2$ Heegaard diagram for the $ 3$-sphere and $ \left\langle {{a_1}, {a_2}; {{\tilde r}_1}, {{\tilde r}_2}} \right\rangle $ be the cyclically reduced presentation associated with $ D$. We shall show that $ {{\tilde{r}}_1}$ contains $ {{\tilde{r}}_2}$ or $ {\tilde{r}}_2^{-1}$ as a subword in cyclic sense if $ \left\{{\tilde r}_1, {\tilde r}_2 \right\} \ne \left\{{a_1}^{\pm 1}, {a_2}^{\pm 1} \right\}$ holds, and that, using this property, $ \left\langle {a_1}, {a_2};{r_1}, {r_2} \right\rangle $ can be transformed to the trivial one $ \left\langle {{a_1}, {a_2};{a_1}^{\pm 1}, a_2^{\pm 1}} \right\rangle $. By the recent positive solution of genus $ 2$ Poincaré conjecture, our result implies the purely algebraic, algorithmic solution to the decision problem; whether a given $ 3$-manifold with a genus $ 2$ Heegaard splitting is simply connected or not, equivalently, is homeomorphic to the $ 3$-sphere or not.

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Keywords: $ 3$-sphere, $ 3$-manifolds, Heegaard diagrams of genus $ 2$, fake Heegaard diagrams, surgeries, wave, group presentations, substitutions, strongly simply trivial
Article copyright: © Copyright 1983 American Mathematical Society