Local detection of strongly irreducible Heegaard splittings
type: Item
instance of: scholarly article
title: Local detection of strongly irreducible Heegaard splittings
zbMATH Open document ID: 0926.57018
DOI: 10.1016/S0166-8641(97)00184-3
published in: Topology and its Applications
publication date: 1999-11-22T00:00:00Z
review text: Let \(M\) be a compact \(3\)-manifold containing a Heegaard surface \(S\) that splits \(M\) into two handlebodies \(H_1\) and \(H_2\) (or, if \(M\) has nonempty boundary, into two compression bodies). \textit{A. J. Casson} and \textit{C. McA. Gordon} [ibid. 27, No. 3, 275-283 (1987; Zbl 0632.57010)] introduced the notion of a strongly irreducible Heegaard splitting. This means that any pair of essential discs, one in \(H_1\) and one in \(H_2\), must have intersecting boundaries. Such splittings are prevalent. For example, in a non-Haken \(3\)-manifold, any splitting either is strongly irreducible or is reducible, and consequently any minimal genus splitting is strongly irreducible. In this paper, the author shows that for a strongly irreducible Heegaard splitting, the possible intersections of \(S\) with a \(3\)-ball or a solid torus are significantly constrained. If \(B\) is a \(3\)-ball in \(M\) (transverse to \(S\)) for which the planar surfaces \(\partial B\cap H_i\) are incompressible, then \(S\cap B\) is connected, planar, and unknotted in \(B\) (the latter means that it is parallel to a submanifold of \(\partial B\)). If \(V\) is a solid torus such that \(\partial V\) intersects \(S\) in parallel essential curves, not bounding discs in \(V\), then \(S\) intersects \(V\) in a collection of boundary-parallel annuli and possibly one other component, obtained from one or two annuli by attaching a tube along an arc parallel to a subarc of \(\partial V\). If the latter sort of component is in \(V\), then \(S-V\) is incompressible in \(M-V\).
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M99
zbMATH DE Number: 1307913
zbMATH Keywords: Heegaard splitting
zbMATH Keywords: strongly irreducible
zbMATH Keywords: solid torus
author: Martin Scharlemann
reviewed by: Darryl McCullough
MaRDI profile type: Publication
cites work: Scindements de Heegaard des espaces lenticulaires
cites work: Reducing Heegaard splittings
cites work: The topological uniqueness of triply periodic minimal surfaces in \(R^ 3\)
cites work: Q5589863
cites work: Heegaard structures of negatively curved 3-manifolds
cites work: Thin position and Heegaard splittings of the 3-sphere
cites work: Q4307578
cites work: Heegaard-Zerlegungen der 3-Sphäre
full work available at URL: s0166-8641(97)00184-3
OpenAlex ID: W2010371484
Recommended article: How a strongly irreducible Heegaard splitting intersects a handlebody
Recommended article: Tori and Heegaard splittings
Recommended article: Local detection of strongly irreducible Heegaard splittings via knot exteriors.
Recommended article: Connected sums of knots and weakly reducible Heegaard splittings
Recommended article: INTERSECTIONS OF CURVES ON SURFACES WITH DISK FAMILIES IN HANDLEBODIES
Recommended article: The disjoint curve property and genus 2 manifolds
Recommended article: Q4253936
Recommended article: Q4539017
Recommended article: Distance of Heegaard splittings of knot complements
Recommended article: CLOSED ESSENTIAL SURFACES AND WEAKLY REDUCIBLE HEEGAARD SPLITTINGS IN MANIFOLDS WITH BOUNDARY
description: scientific article; zbMATH DE number 1307913