The shrinkability of Bing-Whitehead decompositions
This paper studies a special class of toroidal, cell-like decompositions of 3-manifolds. Specifically, the nondegenerate elements consist of the components of \(\cap X_ i\), where generally the ith stage \(X_ i\) of the defining sequence is a locally finite union of pairwise disjont solid tori. The class blends two model construction techniques, one used by \textit{R. H. Bing} [Ann. Math., II. Ser. 56, 354-362 (1952; Zbl 0049.404)] for describing an involution of \(S^ 3\) having a wild 2-sphere as fixed point set, and the other, by \textit{J. H. C. Whitehead} [Q. J. Math. Oxf., II. Ser. 6, 268-279 (1935; Zbl 0013.08103)] for producing a contractible 3- manifold different from \({\mathbb{R}}^ 3\). The former leads to shrinkable decompositions while the latter, by contrast, leads to nonshrinkable ones. The remarkable main result, which pertains to decompositions involving \(b_ 1\) (\(\geq 0)\) states of Bing nested tori, followed by 1 Whitehead nested stage, followed by \(b_ 2\) Bing nested stages, followed by 1 Whitehead nested stage, etc., attests that the associated decomposition is shrinkable if and only if the series \(\sum b_ i/2^ i\) diverges. This clarifies an issue raised in the proof of \textit{M. H. Freedman}'s disk theorem for 4-manifolds [Proc. Int. Congr. Math. Warszawa, Vol. I, 647-663 (1984; Zbl 0577.57003)]. The main result has been proved independently by \textit{D. G. Wright} [Fundam. Math. 132, No. 2, 105-116 (1989; Zbl 0685.57008)].
toroidal, cell-like decompositions of 3-manifolds
scientific article; zbMATH DE number 4113556