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Uniqueness in Waldhausen’s theorem (joint with M. Freedman)

Geometry/Topology

Speaker: Martin Scharlemann, UCSB
Location: 1147 MSB
Start time: Tue, Mar 13 2018, 1:10PM

It is a famous theorem of Waldhausen that any genus g Heegaard splitting surface H in S3 is isotopic to the standard genus g splitting surface H0. Rieck’s modern account of Waldhausen's theorem exploits a theorem of Casson and Gordon on weakly reducible Heegaard splittings. They show that a weak reduction of the splitting leads to a way of breaking up the surface into a connected sum of lower genus splittings, at which point induction can be used.

But how unique is the isotopy of H to H0? To what extent is the isotopy determined by the choice of a weak reduction? We discuss this in the context of a conjecture of Powell about isotopies of H0 to itself.