The following question was
raised by R. H. Bing: “Is it true that if G is a monotone decomposition of E8 into
straight line intervals and one-point sets, then ES∕G is homeomorphic to E3?” In his
paper “Point-like decompositions of E8” he described a possible counter example.
This example has the interesting property that it has many tame cross-sections, but
if its decomposition space is homeomorphic to E8, its set of nondegenerate
elements would have to form a wild Cantor set. This suggests that it would be
interesting to study the connection between the embedding of a cross-section and
the embedding of the set of nondegenerate elements in the decomposition
space.
