Shrinking without lengthening. (English) Zbl 0673.57011

Bing, R. H.

Topology 27, No. 4, 487-493 (1988).

In a famous paper [Ann. Math., II. Ser. 56, 354-362 (1952; Zbl 0049.404)], in which he first began to unravel mysteries concerning the topology of Euclidean 3-space, the author used what we now regard as decomposition theory techniques to produce an involution of the 3-sphere, \(S^ 3\), having a wild 2-sphere as its fixed point set. He reduced the involution problem to showing that a certain related decomposition of \(S^ 3\) reproduced \(S^ 3\), and he solved the latter with an ingenious shrinking argument. Prompted by a question of Michael Freedman, just before his death Bing discovered a new, more efficient shrinking method that does not increase the centerline lengths among the solid tori appearing in the defining sequence, which he sets forth in the paper under review. Freedman’s interest in the matter stems in part from investigations into the existence of uniformly quasiconformal group actions on \(S^ n\) that are not conjugate to conformal actions, the subject of his recent paper with R. Skora [M. H. Freedman and R. Skora, J. Differ. Geom. 25, 75-98 (1987; Zbl 0588.57024)], where connections with this shrinking argument of the author are highlighted.

Reviewer: R.J.Daverman

MSC:

57M30	Wild embeddings
54B15	Quotient spaces, decompositions in general topology
57N10	Topology of general \(3\)-manifolds (MSC2010)
57S25	Groups acting on specific manifolds

Keywords:

Alexander horned balls; tame arcs; shrinkable decomposition; wild 2- sphere as fixed point set; involution of the 3-sphere; decomposition of \(S^ 3\); centerline lengths; uniformly quasiconformal group actions

Citations:

Zbl 0602.57028; Zbl 0049.404; Zbl 0588.57024