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Shrinking of toroidal decomposition spaces

Volume 227 / 2014

Daniel Kasprowski, Mark Powell Fundamenta Mathematicae 227 (2014), 271-296 MSC: Primary 57M25, 57M30, 57N12. DOI: 10.4064/fm227-3-3

Abstract

Given a sequence of oriented links $L^1,L^2,L^3,\dots $ each of which has a distinguished, unknotted component, there is a decomposition space $\mathcal {D}$ of $S^3$ naturally associated to it, which is constructed as the components of the intersection of an infinite sequence of nested solid tori. The Bing and Whitehead continua are simple, well known examples. We give a necessary and sufficient criterion to determine whether $\mathcal {D}$ is shrinkable, generalising previous work of F. Ancel and M. Starbird and others. This criterion can effectively determine, in many cases, whether the quotient map $S^3 \to S^3 / \mathcal {D}$ can be approximated by homeomorphisms.

Authors

  • Daniel KasprowskiWestfälische Universität Münster
    Einsteinstrasse 62
    48149 Münster, Germany
    e-mail
  • Mark PowellDepartment of Mathematics
    Indiana University
    Bloomington, IN 47405, U.S.A.
    and
    Max Planck Institute for Mathematics
    Vivatsgasse 7
    53111 Bonn, Germany
    e-mail

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