The combinatorics of piecewise linear manifolds by colored graphs
type: Item
instance of: scholarly article
zbMATH Open document ID: 1254.57005
publication date: 2012-09-12T00:00:00Z
full work available at URL: index.html
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 57-02
zbMATH DE Number: 6081597
zbMATH Keywords: PL-manifold
zbMATH Keywords: contracted triangulation
zbMATH Keywords: colored graph
zbMATH Keywords: crystallization
zbMATH Keywords: orientability
zbMATH Keywords: connected sum
zbMATH Keywords: homotopy
zbMATH Keywords: homology
zbMATH Keywords: regular genus
zbMATH Keywords: complexity
zbMATH Keywords: average order
author: Alberto Cavicchioli
author: Fulvia Spaggiari
MaRDI profile type: Publication
title: The combinatorics of piecewise linear manifolds by colored graphs
published in: International Mathematical Forum
review text: The present paper deals with crystallization theory, which is an interesting combinatorial representation method for piecewise-linear (PL) manifolds of arbitrary dimension: see \textit{M. Ferri, C. Gagliardi} and \textit{L. Grasselli} [Aequationes Math. 31, 121--141 (1986; Zbl 0623.57012)] for a survey on the foundational aspects of the theory, \textit{M. R. Casali, P. Bandieri} and \textit{C. Gagliardi} [Atti Semin. Mat. Fis. Univ. Modena 49, Suppl., 283--337 (2001; Zbl 1420.57066)] and \textit{P. Bandieri} et al. [ibid. 58, 11--45 (2011; Zbl 1263.57018)] for subsequent improvements.NEWLINENEWLINEThe authors give an overview of classical results and techniques on crystallizations from a graph-theoretical point of view, paying particular attention to certain combinatorial invariants such as regular genus, complexity and average order. Several open problems and conjectures concerning the above invariants are also included in the present survey paper.NEWLINENEWLINEIt should be noted that, for some of the covered topics, the overview of the theory could be completed with significant improvements which are not reported by the authors: see for example \textit{M. R. Casali} and \textit{P. Cristofori} [J. Knot Theory Ramifications 17, No. 5, 579--599 (2008; Zbl 1163.57017)] (resp. \textit{M. R. Casali} [Acta Appl. Math. 54, No. 1, 75--97 (1998; Zbl 0913.57011)] and \textit{P. Bandieri, P. Cristofori} and \textit{C. Gagliardi} [J. Knot Theory Ramifications 18, No. 3, 381--395 (2009; Zbl 1177.57021)]) for the topological classification of orientable (resp. non-orientable) 3-manifolds up to reduced complexity 28, or \textit{M. R. Casali} [JP J. Geom. Topol. 10, No. 1, 41--62 (2010; Zbl 1250.57036)] for the complete identification of minimal 3-manifolds up to genus four. It is also worth noting the existence of known relationships among Matveev's complexity and some of the 3-dimensional invariants considered in the present paper see \textit{P. Bandieri} et al. [op. cit.].
reviewed by: Maria Rita Casali
description: scientific article; zbMATH DE number 6081597