Reducing Heegaard splittings
type: Item
instance of: scholarly article
title: Reducing Heegaard splittings
zbMATH Open document ID: 0632.57010
DOI: 10.1016/0166-8641(87)90092-7
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Let \((W,W')\) be a Heegaard splitting of a closed 3-manifold M with the Heegaard surface \(F=\partial W=\partial W'\). If \((W,W')\) is not strongly irreducible (i.e. there are essential disks (D,\(\partial D)\subset (W,F)\), \((D',\partial D')\subset (W',F)\) such that \(\partial D\cap \partial D'=\emptyset)\) then either the splitting is reducible or M contains an incompressible surface. The main tool of the proof is a mild generalization of a result of Haken: if a closed 3-manifold with a given Heegaard splitting contains an essential 2-sphere, then it contains one which meets the Heegaard surface in a single circle. Another application of the Haken result is a sufficient condition for the addition of 2- handles to a 3-manifold to produce an irreducible manifold with incompressible boundary (for another sufficient condition see the reviewer's paper in Low dimensional topology and Kleinian groups, Symp. Warwick and Durham 1984, Lond. Math. Soc. Lect. Note Ser. 112, 273-285 (1986; Zbl 0621.57005)). Finally the authors answer a question of B. Maskit by showing that if W is a handlebody and the homotopy class of a simple loop in \(\partial W\) is a proper power in \(\pi_ 1(W)\), then it is a proper power of an element which is represented by a simple loop in \(\partial W\).
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 4028425
zbMATH Keywords: Heegaard splitting
zbMATH Keywords: strongly irreducible
zbMATH Keywords: incompressible surface
zbMATH Keywords: essential 2-sphere
zbMATH Keywords: addition of 2-handles
zbMATH Keywords: incompressible boundary
zbMATH Keywords: handlebody
Wikidata QID: Q56442918
author: Andrew J. Casson
author: Cameron McA. Gordon
reviewed by: Józef H. Przytycki
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90092-7
OpenAlex ID: W2061959778
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description: scientific article; zbMATH DE number 4028425