On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three III - Generalized Heegaard splittings and mapping classes

Let $M$ be an orientable, irreducible $3$-manifold admitting a weakly reducible genus three Heegaard splitting as a minimal genus Heegaard splitting. In this article, we prove that if $[f]$, $[g]\in Mod(M)$ give the same correspondence between two isotopy classes of generalized Heegaard splittings consisting of two Heegaard splittings of genus two, say $[\mathbf{H}]\to [{\mathbf{H}}^{\mathrm{\prime }}]$, then there exists a representative $h$ of the difference $[h]=[g]\cdot [f{]}^{-1}$ such that (i) $h$ preserves a suitably chosen embedding of the Heegaard surface $F^{\mathrm{\prime }}$ obtained by amalgamation from ${\mathbf{H}}^{\mathrm{\prime }}$ which is a representative of $[{\mathbf{H}}^{\mathrm{\prime }}]$ and (ii) $h$ sends a uniquely determined weak reducing pair $(V^{\mathrm{\prime }},W^{\mathrm{\prime }})$ of $F^{\mathrm{\prime }}$ into itself up to isotopy. Moreover, for every orientation-preserving automorphism $\tilde{h}$ satisfying the previous conditions (i) and (ii), there exist two elements of $Mod(M)$ giving correspondence $[\mathbf{H}]\to [{\mathbf{H}}^{\mathrm{\prime }}]$ such that $\tilde{h}$ belongs to the isotopy class of the difference between them.