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).

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