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

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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]∈ Mod(M) give the same correspondence between two isotopy classes of generalized Heegaard splittings consisting of two Heegaard splittings of genus two, say [H]→[H'], then there exists a representative h of the difference [h]=[g]·[f]⁻¹ such that (i) h preserves a suitably chosen embedding of the Heegaard surface F' obtained by amalgamation from H' which is a representative of [H'] and (ii) h sends a uniquely determined weak reducing pair (V',W') of F' into itself up to isotopy. Moreover, for every orientation-preserving automorphism h satisfying the previous conditions (i) and (ii), there exist two elements of Mod(M) giving correspondence [H]→[H'] such that h belongs to the isotopy class of the difference between them.