On Cantor Sets in 3‐Manifolds and Branched Coverings

• Published in: Quarterly journal of mathematics 2003-06, Vol.54 (2), p.209-212
• Main Author: Montesinos-Amilibia, J. M.
• Format: Article
• Language: English
• ISSN: 0033-5606; 1464-3847
• DOI: 10.1093/qmath/hag007

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 Δ, for k = 2, 3 (R. P. Osborne, 1969). It is proved that there exist a k‐cell Δ and a 3‐fold branched covering of M over (a subset of) S3 such that (i) C is tamely embedded in the boundary of Δ, (ii) Δ projects homeomorphically onto a k‐cell Δ̂ tamely embedded in S3, and (iii) C is sent onto a tame Cantor set T tamely embedded in the boundary of Δ̂. The argument uses techniques of branched coverings and is independent of Osborne's theorem.