# “Combinatorial” moves between cell complexes

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).

• Your need to be clearer about your notions of equivalence. For example there are two simplical complex structures on the five-sphere which are not PL equivalent (and thus not connected by such moves). See the “double suspension theorem”. Apr 16, 2022 at 7:38
• Ah. So you are asking for a list of things. In particular, you do not have a fixed research level question… Well, I think that this list would be interesting to have. I will edit your question and hopefully some folks will see fit to provide examples. Apr 17, 2022 at 4:14
• Yes, for CW-complexes there are standard moves, like cell cancellations and "sliding one cell over another". This generates the "simple homotopy equivalence" relation, which is a mild strengthening of the homotopy-equivalence relation. Marshall Cohen's "Simple Homotopy Theory" is a good text for this. But most intro algebraic topology textbooks cover at least part of the story. These moves are called "Whitehead Moves". This is what Smale generalized (to handle decompositions) to prove the h-cobordism theorem. Apr 17, 2022 at 4:43
• To add to @RyanBudney's answer, there is an invariant called Whitehead torsion that obstructs whether two homotopy equivalent complexes are actually simple homotopy equivalent (related by standard moves). Apr 17, 2022 at 5:07
• Can you provide appropriate definitions and references for the statement in the first paragraph (concerning final simplicial complexes)? Googling any two of the three phrases “simplicial complex”, “triangle switch” and “barycentric move” doesn't provide much of an explanation of what the two last are. Apr 17, 2022 at 10:13