Classifying combinatorial 4-manifolds up to complexity
type: Item
instance of: scholarly article
title: Classifying combinatorial 4-manifolds up to complexity
zbMATH Open document ID: 1222.57019
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2010-10-01T00:00:00Z
review text: An \((n + 1)\)-colored graph \((G, c)\) is a finite graph \(G\) along with an edge coloring map \(c : E(G) \to \Delta_n = \{i \in \mathbb Z \;|\; 0 \leq i \leq n\}\) by \(n+1\) colors, where \(E(G)\) is the set of edges of \(G\). To \((G, c)\) we can associate a cellular complex \(K(G)\) of dimension \(n\), such that each \(n\)-simplex of \(K(G)\) (with vertices arbitrarily colored by \(\Delta_n\)) corresponds to a vertex of \(G\). If two vertices of \(G\) are joined by an edge colored by \(i \in \Delta_n\) then the corresponding \(n\)-simplexes are attached along the faces opposite to their vertices colored by \(i\) by a color-preserving identification. \(K(G)\) is defined as the quotient space. If \(M\) is a PL \(n\)-manifold and \(K(G)\) is PL-homeomorphic to \(M\) for some \((G, c)\), we say that \((G, c)\) represents \(M\). If \(K(G)\) has exactly \(n+1\) vertices we say that \((G, c)\) is a crystallization of \(M\). By a theorem of \textit{M. Pezzana} [Atti Sem. Mat. Fis. Univ. Modena 23, 269--277 (1974; Zbl 0314.57005)] and [Boll. Unione Mat. Ital. 12, 98--105 (1975; Zbl 0327.57003)] any closed PL \(n\)-manifold admits a crystallization. Then we can define the complexity \(c(M)\) as the minimum number of vertices that occur in the crystallizations of \(M\). Put \(\widetilde c(M) = c(M) - 2\), the reduced complexity of \(M\). This is a finite-to-one PL invariant with values in the non-negative even integers. In the paper under review the authors give a classification of the closed connected PL \(4\)-manifolds of reduced complexity at most 14. Roughly speaking, the main theorem states that \(\widetilde c(S^4) = 0\), \(\widetilde c(\mathbb CP^2) = 6\), \(\widetilde c(S^1 \times S^3) = \widetilde c(S^1 \, \widetilde\times\, S^3) = 8\), \(\widetilde c(S^2 \times S^2) = 12\), \(\widetilde c(\mathbb RP^4) = 14\), and there are no other prime closed 4-manifolds of reduced complexity \(\leq 14\) up to PL homeomorphisms. In order to prove this theorem, the authors cons…
reviewed by: Daniele Zuddas
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 5794610
zbMATH Keywords: 4-manifold
zbMATH Keywords: crystallization
zbMATH Keywords: colored graph
zbMATH Keywords: reduced complexity
author: Alberto Cavicchioli
author: Fulvia Spaggiari
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5794610