Thin position and Heegaard splittings of the 3-sphere
type: Item
instance of: scholarly article
title: Thin position and Heegaard splittings of the 3-sphere
zbMATH Open document ID: 0820.57005
published in: Journal of Differential Geometry
publication date: 1994-07-18T00:00:00Z
review text: The paper gives a clearly presented simple proof of Waldhausen's theorem, that a Heegaard splitting of \(S^ 3\) depends only on its genus. The main technique is to use the spine of one handlebody in the splitting, which is a trivalent graph, and manipulate it using edge slides, while keeping track of compressing discs in the complementary handlebody. In the case of a splitting of \(S^ 3\), the spine is put into a suitably minimal non-degenerate position relative to a height function; a series of lemmas then shows how to find either an unknotted cycle or a split link on the graph after edge slides. This allows a simple reduction of the splitting to one of lower genus, and an inductive proof of Waldhausen's theorem. Besides this application, the authors point to the lemmas as giving new characterizations of graphs in \(S^ 3\) whose complement is boundary-reducible. As with other writings by these authors, the material is very well organized, and the combinatorial arguments follow so smoothly that their full power is not always immediately apparent.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M15
zbMATH DE Number: 590833
zbMATH Keywords: graphs whose complement is boundary-reducible
zbMATH Keywords: Waldhausen's theorem
zbMATH Keywords: Heegaard splitting of \(S^ 3\)
zbMATH Keywords: height function
zbMATH Keywords: unknotted cycle
zbMATH Keywords: split link
author: Martin Scharlemann
author: Abigail Thompson
reviewed by: H. R. Morton
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214454875
OpenAlex ID: W1526584020
Wikidata QID: Q115179795
DOI: 10.4310/JDG/1214454875
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description: scientific article; zbMATH DE number 590833