EDITED:


A pair of finite simplical complexes are equivalent if and only if they are related by a finite sequence of the Pachner moves.


Is there a similar thing on finite cell complexes? That is, are there “related” notions of equivalence and “similar” theorems “reducing” such equivalence to finite sequences “combinatorial” moves? I am interested in any such examples, and I am not bothered if different examples require some side hypotheses (for instance, restricting to regular cell structures).



“Combinatorial” moves between cell complexes
0. If a pair of finite simplicial complexes are PL manifolds, which are additionally PL homeomorphic, then there is a finite sequence of [bisteller flips](https://en.wikipedia.org/wiki/Pachner_moves) taking one to the other. (These are sometimes also called Pachner moves.)
1. [Kirby calculus](https://en.wikipedia.org/wiki/Kirby_calculus) on handle structures of four-manifolds.
2. Collapses and expansions (of CW complexes) generate the relation of [simple homotopy equivalence](https://en.wikipedia.org/wiki/Simple-homotopy_equivalence). (See Ryan's comments above.)


