A nonshrinkable decomposition of 𝑆³ whose nondegenerate elements are contained in a cellular arc

Abstract

A decomposition G of S1 is constructed with the following properties: (1) The set NG of all nondegenerate elements consists of a null sequence of arcs and J-CL(U{ g e NG}) is a simple closed curve. (2) Each arc contained in J is cellular. (3) J is the boundary of a disk Q that is locally flat except at points of J. (4) The decomposition G is not shrinkable; that is, the decomposition space is not homeomorphic to S3.