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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    $\begingroup$ 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”. $\endgroup$
    – Sam Nead
    Apr 16, 2022 at 7:38
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    $\begingroup$ 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. $\endgroup$
    – Sam Nead
    Apr 17, 2022 at 4:14
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    $\begingroup$ 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. $\endgroup$ Apr 17, 2022 at 4:43
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    $\begingroup$ 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). $\endgroup$
    – Jim Conant
    Apr 17, 2022 at 5:07
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    $\begingroup$ 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. $\endgroup$
    – Gro-Tsen
    Apr 17, 2022 at 10:13

1 Answer 1

  1. If a pair of finite simplicial complexes are PL manifolds, which are additionally PL homeomorphic, then there is a finite sequence of bisteller flips taking one to the other. (These are sometimes also called Pachner moves.)

  2. Kirby calculus on handle structures of four-manifolds.

  3. Collapses and expansions (of CW complexes) generate the relation of simple homotopy equivalence. (See Ryan's comments above.)

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    $\begingroup$ Small comment: Kirby calculus is basically an encoding of two features we know about manifolds. 1) Manifolds (and more generally cobordisms) have handle decompositions and 2) We know "manifolds up to cobordism" thus we have a language of how to build manifolds from the generators of the cobordism group, via handle attachments. There is also 3) Cerf theory tells us (in most cases) how one surgery diagram is related to another. So there are analogues of Kirby calculus in all dimensions. This is the "surgery perspective" on manifolds. $\endgroup$ Apr 17, 2022 at 21:49

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