The mapping class groups of reducible Heegaard splittings of genus two

A $3$-manifold which admits a genus-$2$ reducible Heegaard splitting is one of the $3$-sphere, S² × S¹, 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.