Attaching boundary planes to irreducible open 3-manifolds

Abstract

Given any connected, open 3-manifold $U$ having finitely many ends, a non-compact 3-manifold $M$ is constructed having the following properties: the interior of $M$ is homeomorphic to $U$; the boundary of $M$ is the disjoint union of finitely many planes; $M$ is not almost compact; $M$ is eventually end-irreducible; there are no proper, incompressible embeddings of $S^1\times \mathbf{R}$ in $M$; every compact subset of $M$ is contained in a larger compact subset whose complement is anannular; there is a compact subset of $M$ whose complement is ${\mathbf{P}}^2$-irreducible. If $U$ is irreducible it also has the following two properties: every proper, non-trivial plane in $M$ is boundary-parallel; every proper surface in $M$ each component of which has non-empty boundary and is non-compact and simply connected lies in a collar on $\partial M$. This construction can be chosen so that $M$ admits no homeomorphisms which take one boundary plane to another or reverse orientation. For the given $U$ there are uncountably many non-homeomorphic such $M$. Two auxiliary results may be of independent interest. First, general conditions are given under which infinitely many ``trivial'' compact components of the intersection of two proper, non-compact surfaces in an irreducible 3-manifold can be removed by an ambient isotopy. Second, $n$ component tangles in a 3-ball are constructed such that every non-empty union of components of the tangle has hyperbolic exterior.