  Mogami introduced in 1995 a large class of triangulated 3-dimensional
pseudomanifolds, henceforth called "Mogami pseudomanifolds". He proved an
exponential bound for the size of this class in terms of the number of
tetrahedra. The question of whether all 3-balls are Mogami has remained open
since, a positive answer would imply a much-desired exponential upper bound for
the total number of 3-balls (and 3-spheres) with N tetrahedra.
  Here we provide a negative answer: many 3-balls are not Mogami. On the way to
this result, we characterize the Mogami property in terms of nuclei, in the
sense of Collet-Eckmann-Younan: "The only three-dimensional Mogami nucleus is
the tetrahedron".
