Contractible open 3-manifolds which are not covering spaces
type: Item
instance of: scholarly article
title: Contractible open 3-manifolds which are not covering spaces
zbMATH Open document ID: 0658.57007
DOI: 10.1016/0040-9383(88)90005-5
author: S. H. Smith
published in: Topology
publication date: 1988-01-01T00:00:00Z
review text: In 1935 \textit{J. H. C. Whitehead} [Q. J. Math., Oxf. Ser. 6, 268-279 (1935; Zbl 0013.08103)] refuted his own attempt to prove the Poincaré Conjecture by producing an example of a contractible open 3-manifold which is not homeomorphic to Euclidean 3-space \({\mathbb{R}}^ 3\). The manifold is an increasing union of countably many solid tori, so that each is homotopic to a point in the next, but is self-linked so that it is not isotopic to a torus imbedded in a 3-ball. Although their union is contractible, removing the first torus \(V_ 0\) yields a 3-manifold with infinitely-generated fundamental group, so the union cannot be \({\mathbb{R}}^ 3\). Later workers showed that by varying this construction, uncountably many homeomorphism types of contractible open 3-manifolds can be produced. It has been a natural and longstanding question whether such a manifold can be the universal cover of a compact 3-manifold. Since there are only countable many closed 3-manifolds, most of the examples of contractible open 3-manifolds cannot be, but there are no known examples of compact 3-manifolds whose universal covers are contractible and not \({\mathbb{R}}^ 3\). The question became even more interesting when \textit{M. Davis} [Ann. Math., II. Ser. 117, 293-324 (1983; Zbl 0531.57041)] produced, for each \(n\geq 4\), examples of contractible open n-manifolds not homeomorphic to \({\mathbb{R}}^ n\) which are universal covers of compact n-manifolds. In the paper under review, the author proves that a class of contractible open 3-manifolds, which includes the Whithead example, cannot be universal covers of compact 3-manifolds. In fact, he proves the substantially stronger result that they do not admit any nontrivial fixed-point free properly discontinuous group actions (i.e. there cannot be even a single nontrivial covering tranformation). To accomplish this, the author must develop new techniques for working with these examples. He decomposes them into the initial solid torus \(V_ 0\)…
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 57S30
zbMATH DE Number: 4076073
zbMATH Keywords: Whitehead manifold
zbMATH Keywords: contractible open 3-manifold which is not homeomorphic to Euclidean 3-space
zbMATH Keywords: universal cover of a compact 3-manifold
zbMATH Keywords: fixed-point free properly discontinuous group actions
zbMATH Keywords: incompressibility
reviewed by: Darryl McCullough
MaRDI profile type: Publication
full work available at URL: 0040-9383(88)90005-5
OpenAlex ID: W1979115848
description: scientific article; zbMATH DE number 4076073