Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Powell's Conjecture on the Goeritz group of the 3-sphere is stably true


Speaker: Martin Scharlemann, UCSB
Location: 2112 MSB
Start time: Tue, Nov 21 2023, 2:10PM

The genus g Goeritz group G_g of the 3-sphere is (roughly) the fundamental group of the configuration space of genus g Heegaard surfaces in S^3. It can be viewed as a natural 3-dimensional analogue of the braid group in S^2. In 1980 J. Powell proposed that, for any genus g, G_g is generated by 5 elements. This has been confirmed for g < 4. Here we show that, for arbitrary g, Powell's Conjecture is true after a single stabilization.

In rough detail: There is a natural function G_g ->G_{g+1} obtained by just adding a trivial summand near a prescribed point in the Heegaard surface. We show that this natural function carries all of G_g to the subgroup of G_{g+1} generated by Powell's proposed generators.