  This is part 2 of a 3-part article where we provide an $O(n^2)$-algorithm to
produce a surgery presentation of a 3-manifold induced by a gem with a
resolution. In this part we produce a sequence of colored simplicial
2-complexes which are inverses and dual to the sequence of gems produced in the
first part. The refinements of the DPL-faces that keep appearing are
idempotent: the second refinement of a DPL-face is isomorphic to its first
refinement. This fact inhibits exponentiability.
