On some open 3-manifolds that are branched coverings of the Poincaré homology sphere (Q485090)

A closed set \(C\) in a \(3\)-manifold \(M\) is tame if there is a homeomorphism of \(M\) to itself sending \(C\) onto a subcomplex of some locally finite simplicial complex triangulating \(M\). If there is no such homeomorphism, we say that \(C\) is wild. Whitehead constructed a \(1\)-dimensional continuum \(w\), and showed that the complement of \(w\) is a contractible, open \(3\)-manifold \(W\). We notice here the manifold \(W\) is not an open \(3\)-cell, cf. \textit{W. A. Blankinship} and \textit{R. H. Fox} [Proc. Am. Math. Soc. 1, 618--624 (1950; Zbl 0040.25902)]. Blankinship and Fox studied this \(1\)-dimensional continuum \(w\) in \(S^3\) and found out that there exists an epimorphism from \(\pi_1(W\setminus K)\) onto the alternating group \(A_5\), when \(K\) is a tame knot in \(W\). In the paper under review, the author gives a geometrical explanation of this epimorphism by showing that \(-1\) Dehn surgery on \(K\) yields a branched covering over the Poincaré homology sphere. A similar result is also shown in this paper for the complement in \(S^3\) of a Cantor set constructed by Bing.