Mixed Bing–Whitehead Decompositions

Mixed Bing–Whitehead decompositions are a special class of toroidal decompositions of the 3-sphere, defined as the intersection of infinite nested sequences of solid tori. The Bing decomposition and the Whitehead decomposition from previous chapters are both examples of mixed Bing–Whitehead decompositions. In this chapter a precise criterion for when toroidal decompositions shrink is given, in terms of a ‘disc replicating function’. In the case of mixed Bing–Whitehead decomposition, this measures the relative numbers of Bing and Whitehead doubling in the sequence of solid tori in the definition. Mixed Bing–Whitehead decompositions are related to the boundaries of skyscrapers, and the shrinking theorem proved in this chapter will be key to the eventual proof of the disc embedding theorem.