Embedding open 3-manifolds in compact 3-manifolds

Abstract

We consider open 3-maniίolds that are monotone unions of compact 3-manifolds each bounded by a torus. We give necessary and sufficient conditions for embedding such an open 3-manifold in a compact 3-manifold. We also show that if the open 3-manif old embeds in a compact 3-manif old, then it embeds in a compact 3-manifold as the complement of the intersection of a decreasing sequence of solid tori. 1* Introduction and preliminary definitions* Kister and McMillan [5] showed that a particular contractible open 3-manifold does not embed in S 3. Haken [4] then showed that this same open 3-manifold does not embed in any compact 3-manifold. Haken's major tool was his finiteness theorem stating that there is an upper bound on the number of incompressible nonparallel surfaces in a compact 3-manifold. See [4] and §VI of [12]. This theorem is also the major tool in this paper. The spaces and maps that we will consider are intended to be in the piecewise linear category. An n-manifold is a separable metric space such that each point has a neighborhood homeomorphic to an n-ceϊl. A submanifold of an ^-manifold is a subspace that is also an ^-manifold. A fc-manifold F is properly embedded in an ^-manifold N if and only if F is a closed subset of N and F f] dN = dF. A surface is a connected 2-manifold; a planar surface is a surface that can be embedded in a disk. A punctured disk is a planar surface obtained by removing from the interior of a disk the interiors of a finite collection of disjoint subdisks. Similarly a punctured 3-cell is a compact 3-manifold obtained by removing from the interior of a 3-cell the interiors of a finite collection of disjoint 3-cells. Let / denote the unit interval [0, 1]; let It denote the interval [-i, i\. Two properly embedded surfaces F and F' are parallel in a 3manifold JV if and only if there is an embedding of (F x /, dF x /) into (N, dN) such that F is the image of F x {0} and Ff is the image of F x {1}. A collection of properly embedded surfaces is parallel if and only if any two disjoint surfaces in the collection are parallel. The corresponding definitions for parallel simple closed curves in a 2-manifold are similar. A 2-manifold F properly embedded in a 3-manifold N is compressible if and only if either F is a 2-sphere that bounds a 3-cell in