On contractible open 3-manifolds
Contractible open 3-manifolds M have been constructed by Whitehead and others by exhausting M by a countable family of handlebodies, each contained contractibly in its successor. A manifold obtained in such a way is called a Whitehead manifold, resp. a fake Whitehead manifold, if the handlebodies contain fake 3-cells. Theorem 1. A simply connected irreducible open 3-manifold is a Whitehead manifold. Theorem 3. A contractible open 3-manifold that is not a Whitehead manifold is a fake Whitehead manifold. The proof reduces the claim to a result in [\textit{D. R. McMillan}, Bull. Am. Math. Soc. 67, 510-514 (1961; Zbl 0116.408)] by proving that every compact subset of such a manifold is contained in a (fake) handlebody. The author then constructs a fake Whitehead manifold (assuming the existence of fake 3-spheres) which does not contain an infinite locally finite collection of disjoint fake 3-cells. Finally \(M\times {\mathbb{R}}\cong_{P1}{\mathbb{R}}^ 4 \)for contractible open 3- manifolds M is played back to that property for fake 3-cells.
scientific article; zbMATH DE number 4042856
scientific article