Contractible open 3-manifolds which are not covering spaces
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 \(V_ 0\) and the regions \(V_ n- int(V_{n-1})\). Using results from the theory of compact 3-manifolds, he shows how to reselect this decomposition to satisfy certain incompressibility properties. Then a key lemma, called the Long Isotopy Lemma, shows that any homotopy of a nontrivial loop in \(V_ n\) which moves it outside of \(V_{n+1}\) must pass through \(V_ 0\). Using this, a clever geometric argument shows that for every homeomorphism of the manifold there is a compact set which intersects infinitely many of its translates. Since every finite-order homeomorphism must have a fixed point, this shows there is no fixed-point free properly discontinuous action. The paper is well-written. The author points out shortcuts for the reader who does not wish to take the time to understand the technicalities needed for the most general case. The techniques initiated in this paper have since been further refined by the author, yielding more results about larger classes of noncompact 3-manifolds. The reader should perhaps be cautioned that one of the definitions of properly discontinuous group action given in the introduction is slightly misstated: the definition requiring each point to have a neighborhood which intersects one of its translates only if the translation is by an element of the (finite) stabilizer of the point does not imply the definition that each compact set intersects only finitely many of its translates, unless one additionally requires that the quotient space be Hausdorff. Since the author uses the latter definition throughout the rest of the paper, the misstatement has no effect on its content.
contractible open 3-manifold which is not homeomorphic to Euclidean 3-space
fixed-point free properly discontinuous group actions
scientific article; zbMATH DE number 4076073