Shrinking of toroidal decomposition spaces
Volume 227 / 2014
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.
