Shrinking of toroidal decomposition spaces

TL;DR

This paper establishes a necessary and sufficient criterion for the shrinkability of certain decompositions of the 3-sphere derived from sequences of links with unknotted components, generalizing previous criteria and enabling effective determination of when the quotient map can be approximated by homeomorphisms.

Contribution

The paper provides a new, comprehensive criterion for shrinkability of decomposition spaces associated with sequences of links, extending prior results by Ancel, Starbird, and others.

Findings

Provides a complete criterion for shrinkability of these decompositions.

Enables effective determination of when the quotient map is approximable by homeomorphisms.

Generalizes previous work on Bing and Whitehead continua.

Abstract

Given a sequence of oriented links L^1,L^2,L^3,... each of which has a distinguished, unknotted component, there is a decomposition of the 3-sphere 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 such a decomposition 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 which identifies the elements of the decomposition to points can be approximated by homeomorphisms.