Additivity of complexity and Haken's method in the topology of three- dimensional manifolds
type: Item
instance of: scholarly article
title: Additivity of complexity and Haken's method in the topology of three- dimensional manifolds
zbMATH Open document ID: 0697.57006
DOI: 10.1007/BF01056281
published in: Ukrainian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Denote by \({\mathcal M}\) the set of all compact 3-dimensional manifolds (homeomorphic manifolds are considered the same), and by \(\Delta\) the underlying polyhedron of the 1-skeleton of the standard 3-simplex, i.e. a polyhedron homeomorphic to the circle with three radii. A polyhedron P is called almost special if the link of each point of P can be embedded into \(\Delta\). A function d: \({\mathcal M}\to {\mathbb{Z}}_+\) is defined as follows: \(d(M)=k\in {\mathbb{Z}}_+\) if M has an almost special spine which has k vertices and does not have an almost special spine with less than k vertices. It is proved that the number k(M), which is called the complexity of M, has the following properties: (a) \(d^{-1}(k)\) contains finitely many closed irreducible manifolds (finiteness); (b) \(d(M_ 1\#M_ 2)=d(M_ 1)+d(M_ 2)\), where \(M_ 1\#M_ 2\) denotes the connected sum of \(M_ 1\) and \(M_ 2\) (additivity). The proof of (a) is short and based on previous results of the author, while the proof of (b) is based on \textit{W. Haken}'s theory of surfaces [Theorie der Normalflächen. Ein Isotopiekriterium für den Kreisknoten. Acta Math. 105, 245-375 (1961; Zbl 0100.194)].
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M20
zbMATH DE Number: 4143049
zbMATH Keywords: complexity of a 3-manifold
zbMATH Keywords: almost special polyhedron
zbMATH Keywords: compact 3- dimensional manifolds
zbMATH Keywords: 1-skeleton of the standard 3-simplex
zbMATH Keywords: almost special spine
author: Sergei Vladimirovich Matveev
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
cites work: An Imbedding Theorem for Connected 3-Manifolds with Boundary
cites work: Q4774764
cites work: Theorie der Normalflächen. Ein Isotopiekriterium für den Kreisknoten
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description: scientific article; zbMATH DE number 4143049