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

TL;DR

This paper investigates the relationship between automorphisms and generalized Heegaard splittings of genus three in certain 3-manifolds, establishing conditions under which automorphisms preserve specific structures and pairs.

Contribution

It provides a detailed analysis of how automorphisms relate to generalized Heegaard splittings and their weak reducing pairs in genus three 3-manifolds, extending understanding of mapping class actions.

Findings

Automorphisms preserving embeddings of Heegaard surfaces are characterized.

Weak reducing pairs are shown to be preserved up to isotopy under certain automorphisms.

Conditions are established for automorphisms to correspond to elements of the mapping class group.

Abstract

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 $[\mathbf{H}]\to[\mathbf{H}']$, 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 $F'$ obtained by amalgamation from $\mathbf{H}'$ which is a representative of $[\mathbf{H}']$ 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 $\tilde{h}$ satisfying the previous conditions (i) and (ii), there exist two elements of $Mod(M)$ giving correspondence $[\mathbf{H}]\to[\mathbf{H}']$ such that $\tilde{h}$ belongs to the isotopy class of the difference between them.