On genus $2$ Heegaard diagrams for the $3$-sphere

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.