Powell’s conjecture on the Goeritz group of S3 is stably true

Scharlemann, Martin

doi:10.2140/agt.2025.25.3775

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Powell's conjecture on the Goeritz group of $S^3$ is stably true

Martin Scharlemann

Algebraic & Geometric Topology 25 (2025) 3775–3787

DOI: 10.2140/agt.2025.25.3775

Abstract

In 1980, J Powell proposed that, for every genus $g$ , five specific elements suffice to generate the Goeritz group $𝒢_{g}$ of genus $g$ Heegaard splittings of $S^{3}$ . Powell’s conjecture remains undecided for $g \geq 4$ . Let $𝒫_{g} \subset 𝒢_{g}$ denote the subgroup generated by Powell’s elements. Here we show that, for each genus $g$ , the natural function $𝒢_{g} \to 𝒢_{g + 1} ∕ 𝒫_{g + 1}$ is trivial.

Keywords

Goeritz group, Heegaard splitting, Powell conjecture

Mathematical Subject Classification

Primary: 57K35, 57M50, 57M60

References

Publication

Received: 24 January 2023

Revised: 25 October 2023

Accepted: 13 November 2023

Published: 6 October 2025

Authors

Martin Scharlemann
Department of Mathematics UC Santa Barbara Santa Barbara, CA United States

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Volume 25, issue 6 (2025)