The complex of end reductions of a contractible open 3-manifold: constructing 1-dimensional examples (Q2493418)

Given an irreducible, contractible, open 3-manifold \(W\) which is not homeomorphic to \(\mathbb R^3\), there is an associated simplicial complex \(S(W)\), the complex of end reductions of \(W\). Whenever \(W\) covers a 3-manifold \(M\) one has that \(\pi_1(M)\) is isomorphic to a subgroup of the group \(\Aut(S(W))\) of simplicial automorphisms of \(S(W)\). In the present paper the author presents a new method for constructing examples \(W\) with \(S(W)\) isomorphic to a triangulation of \(\mathbb R\). It follows that any 3-manifold \(M\) covered by \(W\) must have \(\pi_1(M)\) infinite cyclic. The author also gives a complete isotopy classification of the end reductions of these \(W\). The method of the paper seems to be used to construct \(\mathbb R^2\)-irreducible Whitehead manifolds which cover 3-manifolds with non-Abelian free fundamental group and can cover only 3-manifolds with free fundamental group.