On reduced complexity of closed piecewise linear 5-manifolds
type: Item
instance of: scholarly article
title: On reduced complexity of closed piecewise linear 5-manifolds
zbMATH Open document ID: 1499.57014
published in: Periodica Mathematica Hungarica
publication date: 2021-11-27T00:00:00Z
review text: Crystallization theory is a representation method for closed PL \(n\)-manifolds \(M\) by means of a particular class of edge-colored graphs called crystallizations. \textit{J. Bracho} and \textit{L. Montejano} [Geom. Dedicata 22, 303--328 (1987; Zbl 0631.57017)] defined the \textit{reduced complexity} \(\tilde c (M) = c(M) - 2\) of \(M\) within crystallization theory (here \(c(M) \ge 2\) denotes the \textit{complexity} of \(M\), the minimum number of \(n\)-simplices of a contracted triangulation of \(M\)) (for example, for a closed connected surface \(M\), \(\tilde c (M) = 4 - 2\chi(M)\) where \(\chi(M)\) denotes the Euler characteristic). Concentrating on dimension 5 in the present paper, the authors classify the closed connected PL (or smooth) 5-manifolds \(M\) up to reduced complexity 20: the only \(M\) with \(\tilde c (M) = 0\) is the 5-sphere \(\mathbb{S}^5\); there are no \(M\) with \(0 < \tilde c (M) < 10\) and exactly two manifolds \(\mathbb{S}^1 \times \mathbb{S}^4\) and the twisted version \(\mathbb{S}^1 \tilde \times \mathbb{S}^4\) with \(\tilde c (M) = 10\), no \(M\) with \(10 < \tilde c (M) < 20\), and finally a few explicitly described manifolds with \(\tilde c (M) = 20\) (among them \(\mathbb{S}^2 \times \mathbb{S}^3\)). ``In particular, this gives a combinatorial characterization of \(\mathbb{S}^2 \times \mathbb{S}^3\) among closed connected spin PL 5-manifolds.'' The paper starts with a short survey on complexities in general, and in particular in dimensions 3 and 4.
Mathematics Subject Classification ID: 57K50
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 7434711
zbMATH Keywords: combinatorial \(5\)-manifolds
zbMATH Keywords: edge-colored graphs
zbMATH Keywords: crystallization-complexity
zbMATH Keywords: classification
zbMATH Keywords: Poincaré conjecture
zbMATH Keywords: homology
zbMATH Keywords: handles
zbMATH Keywords: surgery
author: Alberto Cavicchioli
author: Fulvia Spaggiari
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: s10998-020-00375-6
OpenAlex ID: W3128174551
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DOI: 10.1007/S10998-020-00375-6
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description: scientific article; zbMATH DE number 7434711