In this note we consider the
problem of determining whether the union of cells is nicely embedded in the n-sphere
if each of the cells is nicely embedded. This question is related to many embedding
problems. For instance, the n-dimensional Annulus Conjecture (now known to
be true for n≠4) is a special case. Cantrell and Lacher have shown that an
affirmative answer implies local flatness of certain submanifolds. Also, this
question is related to the conjecture that an embedding of a complex into the
n-sphere which is locally flat on open simplexes is e-tame in codimension
three.
