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

Martin Scharlemann

TL;DR

The paper advances Powell’s conjecture by proving stable triviality of the Powell cosets: for every genus $g$, the natural map from the Goeritz group $\mathcal{G}_g$ to $\mathcal{G}_{g+1}/\mathcal{P}_{g+1}$ is trivial, where $\mathcal{P}_{g}$ is generated by Powell’s moves. It achieves this by (i) refining Powell’s generators and showing $\mathcal{P}_g$ is generated by $D_\theta$, $D_\omega$, and $\{\phi_i\}$, linking them to the $g$-strand braid group; (ii) establishing that the Powell subgroup already captures numerous braid-type and eyeglass moves, and (iii) constructing a stabilization framework $\iota^+$ and a dihedral-tracking map $\Theta_p$ to control how these moves behave under genus augmentation. The core technical thrust demonstrates that, under natural geometric assumptions verified in the paper, every Powell move remains trivial in the stabilized cosets, and a general argument (augmented by a detailed examination of $K_{2,3}$ symmetries) shows the desired stability. This yields a robust, geometrically transparent pathway toward resolving Powell’s conjecture in higher genus and clarifies the structure of Goeritz groups under stabilization.

Abstract

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

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

TL;DR

The paper advances Powell’s conjecture by proving stable triviality of the Powell cosets: for every genus , the natural map from the Goeritz group to is trivial, where is generated by Powell’s moves. It achieves this by (i) refining Powell’s generators and showing is generated by , , and , linking them to the -strand braid group; (ii) establishing that the Powell subgroup already captures numerous braid-type and eyeglass moves, and (iii) constructing a stabilization framework and a dihedral-tracking map to control how these moves behave under genus augmentation. The core technical thrust demonstrates that, under natural geometric assumptions verified in the paper, every Powell move remains trivial in the stabilized cosets, and a general argument (augmented by a detailed examination of symmetries) shows the desired stability. This yields a robust, geometrically transparent pathway toward resolving Powell’s conjecture in higher genus and clarifies the structure of Goeritz groups under stabilization.

Abstract

In 1980 J. Powell proposed that, for every genus , five specific elements suffice to generate the Goeritz group of genus Heegaard splittings of . Powell's Conjecture remains undecided for . Let denote the subgroup generated by Powell's elements. Here we show that, for each genus , the natural function is trivial.
Paper Structure (8 sections, 26 theorems, 30 equations, 19 figures)

This paper contains 8 sections, 26 theorems, 30 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Powell's generators re-expressed
  3. What the Powell group can do
  4. Stabilization and topological conjugacy
  5. Symmetries of $K_{2, 3}$ as a subgroup of $\mathcal{G}_2$
  6. From $\mathcal{G}_2$ to $\mathcal{G}_g$ - first observations
  7. Towards a useful function $\operatorname{Dih}_6 \to \mathcal{P}_{g}\backslash\mathcal{G}_g/\mathcal{P}_g$
  8. Two examples

Key Result

Proposition 2.1

Powell's proposed generators generate $\mathcal{G}_g$ if and only if they generate $\tilde{\mathcal{G}}_g$.

Figures (19)

  • Figure 1: Powell's picture of $T_g$
  • Figure 6: $D_{\eta} = (D_{\omega})^2 \phi_1 \phi_2\phi_{3}$
  • Figure 7:
  • Figure 8: An eyeglass twist
  • Figure 9: A bubble exchange, via a Goeritz move in $\mathcal{G}_2$.
  • ...and 14 more figures

Theorems & Definitions (51)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • ...and 41 more