Cartesian product stabilization of 3-manifolds
type: Item
instance of: scholarly article
title: Cartesian product stabilization of 3-manifolds
zbMATH Open document ID: 1205.57021
author: Witold Rosicki
author: Slawomir Kwasik
published in: Topology and its Applications
publication date: 2010-09-24T00:00:00Z
review text: It was shown by \textit{K. Borsuk} [Fundam. Math. 33, 273--298 (1945; Zbl 0060.40002)] that any closed manifold has at most one decomposition into the Cartesian product of factors of dimension \(\leq 2\). In the paper under review, the authors investigate the uniqueness of Cartesian product decompositions having a \(3\)-manifold as a factor. Let \(M\) and \(N\) be closed, orientable \(3\)-manifolds which are geometric in the sense of Thurston. Under the natural assumption requiring that \(M\) and \(N\) have no decomposition into Cartesian products, the authors prove that if \(M \times X \approx N \times Y\) (i.e. \(M \times X\) and \(N \times Y\) are homeomorphic) then \(M \approx N\) and \(X \approx Y\) unless \(M\) and \(N\) are Seifert fibered manifolds which are surface bundles over \(S^1\) with the fiber surface of genus \(> 1\), and \(X \approx Y \approx T^2 = S^1 \times S^1\). For Cartesian products with the \(n\)-dimensional torus \(T^n\), they prove that if \(M \times T^n \approx N \times T^n\) for some \(n\), then \(M \approx N\) unless \(M\) and \(N\) are Seifert fibered manifolds as above. The authors also consider uniqueness of Cartesian square decomposition. It was shown by \textit{S. Kwasik} and \textit{R. Schultz} [Topology 41, No.~2, 321--340 (2002; Zbl 0991.57020)] that if two lens spaces have isomorphic fundamental groups, then their Cartesian squares are homeomorphic. On the other hand, in the present paper, the authors prove that if \(M\) and \(N\) are Seifert fibered manifolds which are not lens spaces with the same fundamental group, then \(M \times M\) and \(N \times N\) are homeomorphic only if \(M \approx N\).
reviewed by: Kimihiko Motegi
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 5791173
zbMATH Keywords: Cartesian product
zbMATH Keywords: 3-manifold
MaRDI profile type: Publication
full work available at URL: j.topol.2010.07.003
OpenAlex ID: W2085841102
cites work: On the decomposition of manifolds into curves and surfaces
cites work: Q4143495
cites work: Convergence groups and Seifert fibered 3-manifolds
cites work: Q5683461
cites work: Convergence groups are Fuchsian groups
cites work: Q3871581
cites work: On stability of 3-manifolds
cites work: Vanishing of Whitehead torsion in dimension four
cites work: Toral and exponential stabilization for homotopy spherical spaceforms
cites work: All \(\mathbb{Z}_q\) lens spaces have diffeomorphic squares
cites work: Seifert manifolds
cites work: Q5485637
cites work: Three dimensional manifolds, Kleinian groups and hyperbolic geometry
cites work: Q4041788
DOI: 10.1016/J.TOPOL.2010.07.003
Recommended article: On decomposition of polyhedra into a Cartesian product of 1-dimensional and 2-dimensional factors
Recommended article: Decomposing manifolds into Cartesian products
Recommended article: Cartesian powers of 3-manifolds
Recommended article: All \(\mathbb{Z}_q\) lens spaces have diffeomorphic squares
Recommended article: Two Comments Concerning the Uniqueness of Prime Factorizations for 3-Manifolds
Recommended article: On uniqueness of Cartesian products of surfaces with boundary
Recommended article: On uniqueness of decomposition of 4-polyhedron into Cartesian product of the 2-dimensional factors
Recommended article: On stability of 3-manifolds
Recommended article: Q4430812
Recommended article: Some remarks on the uniqueness of decomposition into Cartesian product
description: scientific article; zbMATH DE number 5791173