Shrinking of toroidal decomposition spaces

Daniel Kasprowski¹ ; Mark Powell²

¹ Westfälische Universität Münster Einsteinstrasse 62 48149 Münster, Germany
² Department of Mathematics Indiana University Bloomington, IN 47405, U.S.A. and Max Planck Institute for Mathematics Vivatsgasse 7 53111 Bonn, Germany

Fundamenta Mathematicae, Tome 227 (2014) no. 3, pp. 271-296

Résumé

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.

Détail

DOI : 10.4064/fm227-3-3

Keywords: given sequence oriented links dots each which has distinguished unknotted component there decomposition space mathcal naturally associated which constructed components intersection infinite sequence nested solid tori bing whitehead continua simple known examples necessary sufficient criterion determine whether mathcal shrinkable generalising previous work ancel starbird others criterion effectively determine many cases whether quotient map mathcal approximated homeomorphisms