The Alexander Gored Ball and the Bing Decomposition

Abstract

‘The Alexander Gored Ball and the Bing Decomposition’ provides a concrete and nontrivial application of the tools of decomposition space theory introduced in the previous chapter. The complement of the Alexander horned ball in the 3-sphere is called the Alexander gored ball. This space is described in three distinct ways: as an intersection of 3-balls; as a 3-dimensional grope; and as a decomposition space. Bing’s theorem that the double of the Alexander gored ball is homeomorphic to the 3-sphere is presented. This gives the first example of a truly nontrivial shrink and, moreover, an example of an exotic involution of the 3-sphere.