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Montesinos Amilibia, José María
(2003)
*On Cantor sets in 3-manifolds and branched coverings.*
Quarterly Journal of Mathematics , 54
(2).
pp. 209-212.
ISSN 0033-5606

Official URL: http://0-qjmath.oxfordjournals.org.cisne.sim.ucm.es/content/54/2.toc

## 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.

Item Type: | Article |
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Uncontrolled Keywords: | 3-manifolds |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 22289 |

Deposited On: | 09 Jul 2013 17:09 |

Last Modified: | 12 Dec 2018 15:13 |

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