Starting with a tree of tetrahedra, suppose that you are allowed to recursively glue together two boundary triangles that have nonempty intersection.
You may perform this type of move as many times you want. Let us call "Mogami manifolds" the triangulated 3-manifolds (with or without boundary) that can be obtained this way. Mogami, a quantum physicist, conjectured in 1995 that all triangulated 3-balls are Mogami. This conjecture implied a much more important one, namely, that "there are only exponentially many triangulation of the 3-sphere with N tetrahedra".
We study this Mogami property in relation other notions, like simply-connectedness, shellability, and collapsibility. With a topological trick we show that Mogami's conjecture is false. The more important conjecture remains unfortunately wide open.