Embeddings and immersions of branched covering spaces (Q801330)

Let \(p: M^ n\to S^ n\), \(n=3\) or 4, be a branched covering in the piecewise linear category with branch set a polyhedral knot or link if \(n=3\), or a locally flat link of 2-spheres if \(n=4\). The author proves that there exists a locally flat embedding \(e: M^ n\to S^ n\times S^ 2\), which commutes with projections onto \(S^ n\), in the following cases: The branched covering \(p: M^ n\to S^ n\) is i) tetrahedral, ii) icosahedral, or iii) r-fold dihedral with r odd, and \(r\geq 3\). These results generalize a theorem of \textit{H. M. Hilden} [Pac. J. Math. 78, 139-147 (1978; Zbl 0422.57003] about 3-fold dihedral branched coverings of \(S^ n\), \(n=3\) or 4.