The word problem in a class of non-Haken 3-manifolds
type: Item
instance of: scholarly article
title: The word problem in a class of non-Haken 3-manifolds
zbMATH Open document ID: 0826.57010
DOI: 10.1016/0040-9383(94)90012-4
author: Andrew Skinner
published in: Topology
publication date: 1994-07-18T00:00:00Z
review text: Let \(M\) be a closed \(P^2\)-irreducible, non-Haken 3-manifold that admits a singular incompressible surface (different from a singular \(S^2\) or \(P^2\)). Hass, Rubinstein, and Scott showed that then \(M\) admits a singular incompressible surface \(F\) whose lifts to the universal cover of \(M\) are embedded planes any two of which intersect transversely in a collection of lines. The main result of the paper is that the word problem for \(\pi_1 (M)\) is solvable, provided that the lifts of \(F\) form a family \(\Pi\) satisfying the following properties: any two planes from \(\Pi\) are either disjoint or intersect in a single line and any three planes from \(\Pi\) have at most one point in common. This result is then applied to solve the word problem for \(\pi_1 (M)\) in a class of closed 3-manifolds \(M\) which were not previously known to have solvable word problems.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F10
zbMATH DE Number: 590275
zbMATH Keywords: \(P^ 2\)-irreducible, non-Haken 3-manifold
zbMATH Keywords: singular incompressible surface
zbMATH Keywords: universal cover
zbMATH Keywords: word problem
reviewed by: Wolfgang Heil
MaRDI profile type: Publication
full work available at URL: 0040-9383(94)90012-4
OpenAlex ID: W2080914277
description: scientific article; zbMATH DE number 590275