Cataloguing PL 4-manifolds by gem-complexity
type: Item
instance of: scholarly article
title: Cataloguing PL 4-manifolds by gem-complexity
zbMATH Open document ID: 1332.57020
author: Maria Rita Casali
author: Paola Cristofori
published in: The Electronic Journal of Combinatorics
publication date: 2015-11-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1408.0378
full work available at URL: v22i4p25
review text: The authors describe an algorithm which can determine whether two crystallizations give the same (homeomorphic/PL-homeomorphic) PL-manifold or not. Using their algorithm they prove, amongst others, the following. Let \((\Gamma, \gamma)\) be a crystallization of a closed, connected, PL 4-manifold \(M\) and let \(\nu(\Gamma)\) be the order of \((\Gamma, \gamma)\). (i) If \(\nu(\Gamma) \leq 18\) and \(M\) is handle-free then \(M\) is PL-homeomorphic to \(S^4\), \(\mathbb{CP}^2\), \(S^2\times S^2\), \(\mathbb{CP}^2\#\mathbb{CP}^2\), \(\mathbb{CP}^2\#(-\mathbb{CP}^2)\) or \(\mathbb{RP}^4\). (ii) If \(\nu(\Gamma) =20\) and \(M\) is handle-free then either \(M\) is simply connected or \(M\) is PL-homeomorphic to \(\mathbb{RP}^4\). (iii) If \(\nu(\Gamma)\leq 132\) and \(M\) is simply connected then the second Betti number \(\beta_2(M)\) of \(M\) is at most \((\nu(\Gamma) -2)/6\) and \(M\) is homeomorphic to \(S^4\#(\#_q\mathbb{CP}^2)\#(\#_r(-\mathbb{CP}^2))\) or \(S^4\#(\#_{s}(S^2\times S^2))\) for some \(q, r, s \geq 0\) (with \(q+r = \beta_2(M)=2s\)). The algorithm may be useful to decide whether different known triangulations of a 4-manifold \(M\) give the same PL-structure on \(M\) or not.
reviewed by: Basudeb Datta
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57Q25
zbMATH DE Number: 6513826
zbMATH Keywords: crystallization
zbMATH Keywords: coloured triangulation
describes a project that uses: Library of Triangulations
MaRDI profile type: Publication
arXiv ID: 1408.0378
cites work: Exotic smooth structures on small 4-manifolds with odd signatures
cites work: Reducible smooth structures on 4-manifolds with zero signature
cites work: Non-orientable 3-manifolds of complexity up to 7
cites work: Q4648919
cites work: NON-ORIENTABLE 3-MANIFOLDS ADMITTING COLORED TRIANGULATIONS WITH AT MOST 30 TETRAHEDRA
cites work: A census of genus-two 3-manifolds up to 42 coloured tetrahedra
cites work: Rigid gems in dimension \(n\)
cites work: Minimal crystallizations of 3-manifolds
cites work: Simple crystallizations of 4-manifolds
cites work: Random Discrete Morse Theory and a New Library of Triangulations
cites work: Enumeration of non-orientable 3-manifolds using face-pairing graphs and union-find
cites work: A combinatorial characterization of 4-dimensional handlebodies
cites work: Classifying Pl 5-Manifolds by Regular Genus: The Boundary Case
cites work: Classification of nonorientable 3-manifolds admitting decompositions into \(\leqq 26\) coloured tetrahedra
cites work: Computing Matveev's complexity of non-orientable 3-manifolds via crystallization theory
cites work: Computing Matveev's complexity via crystallization theory: the orientable case
cites work: A CATALOGUE OF ORIENTABLE 3-MANIFOLDS TRIANGULATED BY 30 COLORED TETRAHEDRA
cites work: A note about complexity of Lens spaces
cites work: PL 4-manifolds admitting simple crystallizations: framed links and regular genus
cites work: Complexity computation for compact 3-manifolds via crystallizations and Heegaard diagrams
cites work: Classifying PL 5-Manifolds Up to Regular Genus Seven
cites work: Q3147476
cites work: Q4344803
cites work: A triangulated \(K3\) surface with the minimum number of vertices
cites work: On the Genus of Smooth 4-Manifolds
cites work: Crystallisations of 2-Fold Branched Coverings of S 3
cites work: Crystallisation moves
cites work: The Only Genus Zero n-Manifold is S n
cites work: A graph-theoretical representation of PL-manifolds -- a survey on crystallizations
cites work: Q4040346
cites work: Monopole equation and the \(\frac{11}{8}\)-conjecture
cites work: Extending the Concept of Genus to Dimension N
cites work: On the genus of the complex projective plane
cites work: Q3886512
cites work: Handles in graphs and sphere bundles over \(S^ 1\)
cites work: Q4857345
cites work: BLOBS AND FLIPS ON GEMS
cites work: Algorithmic topology and classification of 3-manifolds
cites work: Q4074117
cites work: Combinatorial Properties of theK3 Surface: Simplicial Blowups and Slicings
cites work: On Simply-Connected 4-Manifolds
description: scientific article