THE MAPPING CLASS GROUPS OF REDUCIBLE HEEGAARD SPLITTINGS OF GENUS TWO

Abstract

A 3-manifold which admits a genus-2 reducible Heegaard splitting is one of the 3-sphere, S-2 x S-1, lens spaces and their connected sums. For each of those manifolds except most lens spaces, the mapping class group of the genus-2 splitting was shown to be finitely presented. In this work, we study the remaining generic lens spaces and show that the mapping class group of the genus-2 Heegaard splitting is finitely presented for any lens space by giving its explicit presentation. As an application, we show that the fundamental groups of the spaces of the genus-2 Heegaard splittings of lens spaces are all finitely presented.