Let
D be any genus 2 Heegaard diagram for the 3-sphere and
be the cyclically reduced presentation associated with
D. We shall show that
contains
or
as a subword in cyclic sense if
holds, and that, using this property, $\langle a_1,
... [Show full abstract] a_2; r_1, r_2\rangle\langle a_1, a_2; a^{\pm 1}_1, a^{\pm 1}_2\rangle$. 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.