Contractible 3-manifolds and the double 3-space property

TL;DR

This paper demonstrates the existence of uncountably many contractible 3-manifolds with and without the double 3-space property, extending Gabai's work on the Whitehead manifold through generalizations of the Whitehead Link.

Contribution

It introduces new families of contractible 3-manifolds with the double 3-space property and those that lack it, expanding understanding of their topological diversity.

Findings

Uncountably many contractible 3-manifolds have the double 3-space property.

Uncountably many contractible 3-manifolds do not have this property.

Extension of interlacing theory to analyze these manifolds.

Abstract

Gabai showed that the Whitehead manifold is the union of two submanifolds each of which is homeomorphic to $\mathbb R^3$ and whose intersection is again homeomorphic to $\mathbb R^3$. Using a family of generalizations of the Whitehead Link, we show that there are uncountably many contractible 3-manifolds with this double 3-space property. Using a separate family of generalizations of the Whitehead Link and using an extension of interlacing theory, we also show that there are uncountably many contractible 3-manifolds that fail to have this property.