Heegaard diagrams and homotopy 3-spheres
type: Item
instance of: scholarly article
title: Heegaard diagrams and homotopy 3-spheres
zbMATH Open document ID: 0646.57006
DOI: 10.1016/0040-9383(88)90033-X
author: Eduardo Rêgo
author: Colin Rourke
published in: Topology
publication date: 1988-01-01T00:00:00Z
review text: This paper suggests a procedure for constructing homotopy 3-spheres via a computer search. In order to state the main result used in this procedure, let \(H_{r+s}\) denote a genus \(r+s\), 3-dimensional handlebody, which is assumed to be standardly embedded in \({\mathbb{R}}^ 3\). The collection of embedded curves \(\{\alpha_ 1,...,\alpha_ r,\beta_ 1,...,\beta_ s\}\subset \partial H_{r+s}\) is called an H-system provided (i) this collection does not separate \(\partial H_{r+s}\), (ii) there exist disjoint, compact, orientable surfaces \(F_ 1,...,F_ r\) in \({\mathbb{R}}^ 3-int H_{r+s}\) such that \(\partial F_ i=\alpha_ i\), and (iii) there exist disjointly embedded 2-disks \(D_ 1,...,D_ s\) in \(H_{r+s}\) such that \(\partial D_ i=\beta_ i.\) Theorem: Corresponding to each H-system is a homotopy 3-sphere. Moreover, every homotopy 3-sphere occurs in this way. Briefly, if \(\{\alpha_ 1,...,\alpha_ r,\beta_ 1,...,\beta_ s\}\) is an H-system, by cutting \(H_{r+s}\) along the \(D_ i's\) and adding 2-handles along the \(\alpha_ i's\), one obtains a homotopy 3-cell. Simple algorithms are given for determining when a collection of curves \(\{\alpha_ i,...,\alpha_ r,\beta_ 1,...,\beta_ s\}\subset \partial H_{r+s}\) satisfy conditions (ii) and (iii) - thereby making a computer search viable.
Mathematics Subject Classification ID: 57M40
Mathematics Subject Classification ID: 57N12
Mathematics Subject Classification ID: 57-04
zbMATH DE Number: 4054612
zbMATH Keywords: Heegaard splittings of 3-manifolds
zbMATH Keywords: homotopy 3-spheres via a computer search
zbMATH Keywords: 3-dimensional handlebody
MaRDI profile type: Publication
full work available at URL: 0040-9383(88)90033-x
OpenAlex ID: W2114190730
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description: scientific article; zbMATH DE number 4054612