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Any two isomorphic simplicial complexes are simple-homotopy-equivalent.

This is a fairly simple result, but it is not obvious. Yet I have been surprisingly unable to find it in the literature on discrete Morse theory (e.g., Scoville or Kozlov).

To clarify the definitions involved: Simplicial complexes are assumed to be finite by definition here. Simple-homotopy equivalence is the equivalence relation on the class of all simplicial complexes generated by elementary collapses. That is, we say that two simplicial complexes are simple-homotopy-equivalent if they can be transformed into one another via a chain of elementary collapses or expansions (the inverse of simplicial collapses). Introducing new vertices or removing ghost vertices is allowed in the process, but renaming existing vertices is not. Nevertheless, the above result says that it is possible to rename a vertex by a sequence of elementary collapses and expansions.

The proof idea is to rename vertices one by one (possibly using some "buffer elements" if a new vertex clashes with an old one). Thus, we only need to prove the following fact: If $\Delta$ is a simplicial complex, if $a$ is a vertex of $\Delta$, and if $b$ is not a vertex of $\Delta$, then the complex $\Delta\left[a\mapsto b\right]$ (obtained by replacing $a$ by $b$ in $\Delta$) is simple-homotopy-equivalent to $\Delta$. To show this fact, we introduce the larger complex $\Gamma$ whose vertices are the vertices of $\Delta$ as well as $b$, and whose faces are the faces of $\Delta$ as well as the sets $S \cup \left\{b\right\}$ and $\left(S \setminus \left\{a\right\}\right) \cup \left\{b\right\}$ for all faces $S$ of $\Delta$ that contain $a$. Now, it is easy to see that $\Gamma$ collapses onto $\Delta$ (since we can collapse the new face-pair $S \cup \left\{b\right\}$ and $\left(S \setminus \left\{a\right\}\right) \cup \left\{b\right\}$ for each $S$, starting with the highest-dimensional ones), and likewise $\Gamma$ collapses onto $\Delta\left[a\mapsto b\right]$ as well. Thus, $\Delta\left[a\mapsto b\right]$ is simple-homotopy-equivalent to $\Delta$.

But I would rather not elaborate on this argument in a paper about something else, and would much prefer to have a reference to cite. Note that I am working with simplicial complexes -- not CW complexes, nor spaces -- and it is not obvious to me that results from the other settings can be converted in any automatic way.

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  • $\begingroup$ Did you mean to allow infinite simplicial complexes? In this case the theory is more complicated. Any PL homeomorphism of finite simplicial complexes has zero torsion as is shown e.g. in 25.3 of Cohen's book "A course in simple homotopy theory". More generally, the book outlines Chapman's proof that any homeomorphism of finite CW complexes has zero torsion. $\endgroup$
    – Igor Belegradek
    Commented May 22 at 20:13
  • $\begingroup$ @IgorBelegradek: Oops, I should have said "finite" everywhere. Fixing this. $\endgroup$
    – darij grinberg
    Commented May 22 at 20:13
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    $\begingroup$ I don't think this question is about the deep result that a homomorphism between (the realisations of) two simplicial complexes is a simple equivalence, i.e. that the notions of simple equivalence and whitehead torsion doesn't depend on the chosen triangulation. Instead, it's about checking that if one defines elementary expansions of simplicial complexes as literal inclusions, one gets a notion of simple homotopy equivalence which is invariant under isomorphisms. $\endgroup$
    – Achim Krause
    Commented May 22 at 21:22
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    $\begingroup$ Personally, I find it more natural to define a morphism to be an elementary expansion if it is a pushout along a "horn inclusion", the resulting notion of elementary expanse directly includes isomorphisms of simplicial complexes among the simple homotopy equivalences. But I suppose the thing you're asking for can be done as well. (This reminds me a bit of leaving out the isomorphism in the definition of cobordisms, which leads to unnecessary trouble.) $\endgroup$
    – Achim Krause
    Commented May 22 at 21:26
  • $\begingroup$ @AchimKrause: I'm wondering, does this work for simplicial sets as well? One other thing I'm interested in seeing is a discrete Morse theory for simplicial sets. $\endgroup$
    – darij grinberg
    Commented May 22 at 21:32

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