Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two

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Let H 2 {\mathcal H_2} be the group of isotopy classes of orientation-preserving homeomorphisms of S 3 \mathbb S^3 that preserve a Heegaard splitting of genus two. In this paper, we construct a tree in the barycentric subdivision of the disk complex of a handlebody of the splitting to obtain a finite presentation of H 2 {\mathcal H_2} .