On Cantor sets in 3-manifolds and branched coverings

It is known that any Cantor set C in a 3-manifold M (open or closed) is tamely embedded in the boundary of a k-cell Delta, for k = 2, 3 (R. P. Osborne, 1969). It is proved that there exist a k-cell Delta and a 3-fold branched covering of M over (a subset of) S-3 such that (i) C is tamely embedded in the boundary of Delta, (ii) Delta projects homeomorphically onto a k-cell Delta tamely embedded in S-3, and (iii) C is sent onto a tame Cantor set T tamely embedded in the boundary of Delta . The argument uses techniques of branched coverings and is independent of Osborne's theorem.