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

TL;DR

The paper proves that Powell's Conjecture, which suggests five elements generate the Goeritz group for all genus, is stably true by showing a certain natural function is trivial for all genera.

Contribution

It demonstrates that Powell's Conjecture holds in a stable sense by analyzing the relationship between Goeritz groups across genera.

Findings

The natural map from G_g to G_{g+1}/P_{g+1} is trivial for all g.

Powell's Conjecture is stably true across all genera.

Provides new insights into the structure of the Goeritz group and Powell's elements.

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.