On some open 3-manifolds that are branched coverings of the Poincaré homology sphere

Abstract

Blankinship and Fox proved that the Whitehead’s contractible, open \(3\)-manifold \(W\) is not an open \(3\)-cell by constructing a representation \(\rho \) of \(\pi _{1}(W\backslash \alpha )\) into the symmetric group of \(5\) elements \(\Sigma _{5}\), where \(\alpha \) is a particular tame knot in \(W\). A geometric explanation of this fact is given by showing that \(-1\) Dehn surgery on \(\alpha \) is a branched covering over the Poincare homology sphere. Similar constructions exist for the complement in \(S^{3}\) of a Cantor set constructed by Bing, leading to the conjectured existence of a branched covering \(p:M\rightarrow S^{3}\) of a \(3\)-manifold \(M\), branched over a (wildly embedded) Cantor set.