Uniqueness in Haken's Theorem

Michael Freedman; Martin Scharlemann

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arXiv · 2004.07385

Uniqueness in Haken's Theorem

Michael Freedman·Martin Scharlemann

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Abstract

Following Haken and Casson-Gordon, it was shown in [Sc] that given a reducing sphere or boundary-reducing disk E in a Heegaard split manifold M, the Heegaard surface T can be isotoped so that it intersects E in a single circle. Here we show that when this is achieved by two different positionings of T, one can be moved to the other by a sequence of 1) isotopies of T rel E 2) pushing a stabilizing pair of T through E and 3) eyegelass twists of T. The last move is inspired by one of Powell's proposed generators for the Goeritz group.

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Michael Freedman, Martin Scharlemann. 2020-04-15. Uniqueness in Haken's Theorem. https://arxiv.org/abs/2004.07385

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