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This is part 2 of a 3-part article where we provide an $O(n^2)$-algorithm toproduce a surgery presentation of a 3-manifold induced by a gem with aresolution. In this part we produce a sequence of colored simplicial2-complexes which are inverses and dual to the sequence of gems produced in thefirst part. The refinements of the DPL-faces that keep appearing areidempotent: the second refinement of a DPL-face is isomorphic to its firstrefinement. This fact inhibits exponentiability.

Framed link presentations of 3-manifolds by an $O(n^2)$ algorithm, II: colored complexes and boundings in their complexity