# Mathematics Colloquia and Seminars

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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 S

^{3}is isotopic to the standard genus g splitting surface H_{0}. 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 H_{0}? 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 H_{0 }to itself.