Bing’s Papers on Decomposition Spaces, 1957–1959

"… People were interested in knowing whether there was an involution of a 3-sphere onto itself different from ordinary involutions. There was a suspicion that probably the union of two solid Alexander horned spheres sewed together along their boundary gave S3 and that an involution that interchanged these two solid spheres (or Alexander balls) would be an involution equivalent to a standard involution. By the use of strings, rubber bands, and other methods, I was able to show that the union of two solid horned spheres sewed together along their boundaries with the identity homeomorphism was indeed a 3-sphere."


 "It was Bing’s method of proof that was both startling and seminal. He used 'decomposition theory.' The method that Bing used involved a procedure whereby the decomposition elements were gradually 'shrunk' to smaller sets without allowing the other decomposition elements to grow too large. This technique grew into what McAuley later named 'Bing’s shrinking criterion.'

When it turned out that the decomposition space was S3, Bing immediately started trying to frustrate this outcome, that is, he tried to frustrate the shrinking trick. The result of his efforts was perhaps Bing’s greatest construction, the ‘dogbone space’.” (Brown)

Bing’s Dogbone Space is a quotient space of the standard three-dimensional Euclidean space (R3). The space is made by “gluing together” pieces of the space in an iterative method. At one of the stages the space resembles a dog bone. The interesting part is that the resulting space is not homeomorphic to (does not have all the same properties as) the original space, showing that the two dimensional space R2 and R3, despite being very similar, behave quite differently, for such a decomposition in R2 is always equivalent to the original space. In 1959, Bing demonstrated that if one took the product of his Dogbone Space with a copy of R1, then one would get something equivalent to R4 back.