Homeomorphisms of 𝑆³ leaving a Heegaard surface invariant

Powell, Jerome

doi:10.1090/S0002-9947-1980-0549161-6

Homeomorphisms of $S^{3}$ leaving a Heegaard surface invariant
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Trans. Amer. Math. Soc. 257 (1980), 193-216 Request permission

Abstract:

We find a finite set of generators for the group ${\mathcal {H} _g}$ of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere ${S^3}$ which leave a Heegaard surface T of genus g in ${S^3}$ invariant. We also show that every element of the group ${\mathcal {H} _g}$ can be represented by a deformation of the surface T in ${S^3}$ 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."

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