A “rearrangement” theorem of
Wallace states essentially that if a manifold M is the trace of a sequence of spherical
modifications of various types then these modifications can be arranged
so that the order in which they are performed is that of increasing type,
their trace still being M. In this paper a related rearrangement problem
is considered; namely, to determine bounds on how “mixed” the order of
performing a sequence of modifications can be and still possess the same trace
M.