On small geometric invariants of 3-manifolds
type: Item
instance of: scholarly article
title: On small geometric invariants of 3-manifolds
zbMATH Open document ID: 1223.57006
author: Paul J. Kapitza
published in: The New York Journal of Mathematics
publication date: 2011-08-02T00:00:00Z
full work available at URL: https://eudml.org/doc/227465
full work available at URL: 17-18.html
review text: If the \(2\)-spine of a \(3\)-manifold \(M\) has \(k\) edges, the corresponding Heegaard decomposition has \(k\) handles. The corresponding presentation of the fundamental group has \(k\) generators. It was conjectured that if the minimal genus Heegard decomposition of \(M\) was \(k\), then any presentation of the fundamental group of \(M\) would require at least \(k\) generators. \textit{M. Boileau} and \textit{H. Zieschang} showed this was not true. In [Invent. Math. 76, 455--468 (1984; Zbl 0538.57004)], they described a class of counterexamples. The author focuses attention on one of these counterexamples: a \(3\)-manifold \(M_1\) with fundamental group having a finite presentation with \(2\) generators despite \(M_1\) having Heegaard genus \(3\). In the absence of a spine of \(M_1\) corresponding to that \(2\) generator presentation, following Craggs one may seek an appropriate \(2\)-complex in \((M_1-(\text{3-cell})) \times [-1,1] \) carrying a presentation with \(2\) generators for the fundamental group of \(M_1\). The last 24 pages of the paper contain over 400 (mostly simple) equations and 17 increasingly detailed and impressive (and not so simple) diagrams which together describe the construction of such a \(2\) generator complex in \((M_1-(\text{3-cell})) \times [-1,1] \) from a \(3\) generator spine of \(M_1\).
reviewed by: Lee P. Neuwirth
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57R65
zbMATH DE Number: 5935412
zbMATH Keywords: Heegaard genus
zbMATH Keywords: extended Nielsen genus
zbMATH Keywords: spine
zbMATH Keywords: fundamental group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5935412