On Cantor sets in 3-manifolds and branched coverings

Abstract

In 1969 R. P. Osborne [Fund. Math. 65 (1969), 147–151;] proved that any Cantor set in an n-manifold (open or closed) is tamely embedded in the boundary of a k-cell, for every 2≤k≤n. In the present work the author generalizes Osborne's result in the particular case where the manifold has dimension 3. Namely, he proves the following: Theorem 2. Let C be a Cantor set in an orientable 3-manifold M (open or closed). Then there exist a (possibly empty) 0-dimensional subset R of S3, a k-cell Δk⊂M (k=2,3), and a 3-fold covering p:M→S3−R, branched over a locally finite disjoint union of strings, such that (i) C is tamely embedded in the boundary of Δk, (ii) p|Δk is a homeomorphism onto its image, (iii) p(Δk) is a tamely embedded k-cell in S3−R, and (iv) p(C) is a tame Cantor set T in S3−R tamely embedded in the boundary of p(Δk). The proof is based on previous work of the author on branching coverings of 3-manifolds.