Homeomorphisms of 𝑆³ leaving a Heegaard surface invariant

Abstract

We find a finite set of generators for the group H g {\mathcal {H} _g} of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere S 3 {S^3} which leave a Heegaard surface T of genus g in S 3 {S^3} invariant. We also show that every element of the group H g {\mathcal {H} _g} can be represented by a deformation of the surface T in S 3 {S^3} of a very special type: during the deformation the surface T is the boundary of the regular neighborhood of a graph embedded in a fixed 2-sphere. The only exception occurs when a subset of the graph contained in a disc on the 2-sphere is “flipped over."