Cartesian powers of 3-manifolds
type: Item
instance of: scholarly article
title: Cartesian powers of 3-manifolds
zbMATH Open document ID: 1105.57018
author: Reinhard Schultz
author: Slawomir Kwasik
published in: Topology and its Applications
publication date: 2006-12-14T00:00:00Z
review text: In a previous paper [Topology 41, 321--340 (2002; Zbl 0991.57020)], the authors showed that two 3-dimensional lens spaces \(L\) and \(L^{\prime }\) with isomorphic fundamental groups have diffeomorphic Cartesian squares; i.e., one has \( L\times L\sim L^{\prime }\times L^{\prime }.\) It should be noted that \(L\) and \(L^{\prime }\) do not even have to be homotopy equivalent. The question on the existence of spaces with this property was originally formulated by \textit{S. Ulam} in 1933, and as indicated by the authors [op. cit.], lens spaces are the lowest dimensional compact examples that one can produce. It is natural to ask what other sorts of 3-dimensional examples of this sort exist. Examples of noncompact (open) 3-manifolds \(W^{3}\) such that \(W^{3}\nsim \mathbb{R}^{3},\) but \(W^{3}\times W^{3}=\mathbb{R}^{6}\) have been known for some time [cf. \textit{J. Glimm}, Bull. Soc. Math. Fr. 88, 131--135 (1960; Zbl 0094.36101); \textit{K. W. Kwun}, Ann. Math. 79, 104--108 (1964; Zbl 0119.38602) or \textit{D. R. McMillan Jr.}, Bull. Am. Math. Soc. 67, 510--514 (1961; Zbl 0116.40802)]; one such example is the Whitehead 3-manifold that is contractible but not simply connected at infinity [\textit{J. H. C. Whitehead}, Quart. J. Math. 6, 268--279 (1935; JFM 61.0607.01)]. These suggest that one should concentrate on the existence of compact 3-manifolds \(A,B\) such that \(A\times A\) is homeomorphic to \( B\times B\) but \(A\) is not homeomorphic to \(B.\) The purpose of the present paper is to show that no examples of this sort exist in a large class of basic 3-manifolds whose fundamental groups are either infinite or finite but noncyclic. A topological pair \(X,Y\) is called exponentially stable if \( \prod^{n}\) \(X\approx \prod^{n}\) \(Y\) (here \(\approx \) stands for homeomorphic). The authors prove the following main results. Theorem 2. Let \(M\) and \(N\) be lens spaces with isomorphic fundamental groups. Then the pair \(M,\) \(N\) is NOT exponentially stable i…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57N15
zbMATH DE Number: 5080163
zbMATH Keywords: irreducible geometric 3-manifold
zbMATH Keywords: Cartesian power
zbMATH Keywords: Hopfian group
zbMATH Keywords: Large Seifert manifold
zbMATH Keywords: Lyndon spectral sequence
zbMATH Keywords: Gromov norm
zbMATH Keywords: Prime decomposition
zbMATH Keywords: Kneser conjecture
MaRDI profile type: Publication
full work available at URL: j.topol.2006.04.006
OpenAlex ID: W2060613051
cites work: Seminar on Transformation Groups. (AM-46)
cites work: Q5658847
cites work: Q5649690
cites work: Convergence groups and Seifert fibered 3-manifolds
cites work: Q5676841
cites work: Solvable Fundamental Groups of Compact 3-Manifolds
cites work: Homotopy hyperbolic 3-manifolds are hyperbolic
cites work: Two cartesian products which are euclidean spaces
cites work: Q3253828
cites work: Every finite complex has the homology of a duality group
cites work: Q4114528
cites work: Q5289023
cites work: Scindement d'une équivalence d'homotopie en dimension $3$
cites work: Q5648522
cites work: Homotopy equivalences of 3-manifolds with boundaries
cites work: Q3924842
cites work: Product of Euclidean spaces modulo an arc
cites work: Vanishing of Whitehead torsion in dimension four
cites work: All \(\mathbb{Z}_q\) lens spaces have diffeomorphic squares
cites work: Seifert manifolds
cites work: Cartesian products of contractible open manifolds
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
cites work: Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms
cites work: Strong Rigidity of Locally Symmetric Spaces. (AM-78)
cites work: Smooth structures on \(S^ p \times S^ q\)
cites work: There are no fake Seifert fibre spaces with infinite \(\pi_1\).
cites work: A rigidity theorem for Haken manifolds
cites work: Projective planes in irreducible 3-manifolds
cites work: Three dimensional manifolds, Kleinian groups and hyperbolic geometry
cites work: On irreducible 3-manifolds which are sufficiently large
cites work: Q4256667
cites work: The \(\pi_ 1\)-injectivity of self-maps of nonzero degree on 3-manifolds
cites work: A CERTAIN OPEN MANIFOLD WHOSE GROUP IS UNITY
cites work: Surfaces and planar discontinuous groups. Revised and expanded transl. from the German by J. Stillwell
DOI: 10.1016/J.TOPOL.2006.04.006
Recommended article: All \(\mathbb{Z}_q\) lens spaces have diffeomorphic squares
Recommended article: Cobordisms and Reidemeister torsions of homotopy lens spaces
Recommended article: Cartesian product stabilization of 3-manifolds
Recommended article: Tangential thickness of manifolds
Recommended article: Über den Homotopietyp von Linsenraumprodukten
Recommended article: The reducibility of surgered \(3\)-manifolds and homology \(3\)-spheres
Recommended article: Q3651415
Recommended article: Contractible open 3-manifolds which are not covering spaces
Recommended article: Groups with isomorphic homology groups and non-cancellation in homotopy theory
Recommended article: Homotopieäquivalenzen zwischen Produkten aus dreidimensionalen Linsenräumen
description: scientific article; zbMATH DE number 5080163