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Generating the Goeritz group


Speaker: Martin Scharlemann, UCSB
Location: Zoom
Start time: Tue, Oct 26 2021, 4:10PM

Let $T_0$ be the standard genus g Heegaard surface in the 3-sphere. In 1968 Waldhausen showed that any genus g Heegaard surface T in $S^3$ is isotopic to $T_0$. But there could be many choices for such an isotopy. How do they differ?

Put another way, for T a genus g surface, is there a natural set of generators for $\pi_1(Imb(T, S^3), T_0)$, that is loops of embeddings of T in $S^3$ that begin and end with the standard embedding? Doing a half-twist on a standard summand is an example of a non-trivial element of the group. So is exchanging two standard summands.

In 1980 J. Powell proposed (indeed he thought he had proven) that five specific elements sufficed (one of the five has since turned out to be redundant). Powell's conjecture is known to be true for genus 1, 2, and 3. Here we prove that for higher genus

  1. an expansion of Powell's proposed generators to allow topological conjugacy and to include all eyeglass twists does suffice and,
  2. as a consequence, Powell’s Conjecture is stably true.

The theorem is an outgrowth of joint work with Michael Freedman; it is a sprawling piece of work with not yet all its pieces nailed down in e-print, but I hope to convey at least the flavor of the best bits.