Some contractible open manifolds and coverings of manifolds in dimension three

Abstract

McMillan has shown that every irreducible, contractible, open 3-manifold is the monotone union of handlebodies (only 0- and 1-handles) and that there are uncountably many such manifolds. Work by Myers and Wright shows that no irreducible, contractible, open 3-manifold different from R 3 can nontrivially cover any 3-manifold when the handlebodies all have genus one or have bounded genus. We describe a family of irreducible, contractible, open 3-manifolds that we call composite Whitehead manifolds. These manifolds have the property that when written as the monotone union of handlebodies, the handlebodies must have unbounded genus. We show that there are uncountably many composite Whitehead manifolds that nontrivially cover open 3-manifolds but do not cover a compact 3-manifold. We also show that there exist uncountably many composite Whitehead manifolds which cannot nontrivially cover any 3-manifold. It is a famous unsolved problem if any irreducible, contractible, open 3-manifold different from R 3 can cover a compact 3-manifold. It is unlikely that any composite Whitehead manifold covers a compact manifold, but our techniques are not strong enough to answer this question.