All orientable 2-manifolds have finitely many minimal triangulations
type: Item
instance of: scholarly article
title: All orientable 2-manifolds have finitely many minimal triangulations
zbMATH Open document ID: 0654.57010
DOI: 10.1007/BF02767355
author: D. W. Barnette
author: Allan L. Edelson
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The boundary of a tetrahedron is a triangulation of the 2-sphere \(S_ 2\) that is minimal in the sense that no edge can be shrunk to a single vertex. A classical result of Steinitz [\textit{E. Steinitz}, Vorlesungen über die Theorie der Polyeder under Einschluß der Elemente der Topologie. Aus dem Nachlaß hrsg. u. erg. v. Hans Rademacher (1934; Zbl 0009.36503)] asserts that every triangulation of \(S_ 2\) can be reduced to this one by shrinking edges, i.e. there are no other minimal triangulations of \(S_ 2\). The number of minimal triangulations of the projective plane and the torus are known to be 2 and 22, respectively. The authors show that this number is finite for every compact orientable 2-manifold. The proof is by induction on the genus, using the following ideas. Every edge of a minimal triangulation of \(M\neq S_ 2\) belongs to a 3-circuit that does not bound a cell, and the number of such circuits passing through a given vertex is bounded by a constant depending only on the genus g of M. This allows to produce from any minimal triangulation of \(M_ g\) a minimal triangulation of \(M_{g-1}\) using surgery and a bounded number of edge shrinkings.
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 52Bxx
zbMATH DE Number: 4067881
zbMATH Keywords: minimal triangulations
zbMATH Keywords: compact orientable 2-manifold
zbMATH Keywords: surgery
zbMATH Keywords: edge shrinkings
MaRDI profile type: Publication
cites work: Generating the triangulations of the projective plane
cites work: Q3766517
cites work: Q4089424
full work available at URL: https://doi.org/10.1007/bf02767355
OpenAlex ID: W2062597318
description: scientific article; zbMATH DE number 4067881