I recently started reading up on simple homotopy theory. Here is a question I stumbled upon.
In his 1938 Paper Simplicial Spaces, Nuclei and m-Groups Whitehead introduced the notion of elementary expansions and elementary collapses of simplicial complexes. Essentially a simplicial complex $K'$ is obtained from $K$ through an elementary collapse, by removing two simplices $p$ and $q$ from $K$ such that:
- $p$ is a maximal simplex of $K$.
- $q < p$ is a (maximal, proper) free face of $p$ (i.e. not contained in any other simplex but itself and $p$).
An elementary expansion is the obvious inverse operation. There are topological realizations of these operations, with the collapse given by "pushing the free face onto the others". Lets call a map of simplicial complexes a simplicial simple homotopy equivalence, if it is homotopic to the composition of such maps.
Later on he decided that CW-complexes were a more convenient setting to work in (Simple Homotopy Types), and this was where he developed the now famous results on whitehead torsion. This is the setting that most of the following material (such as Cohens Book A course in simple homotopy theory) are presented in and probably most familiar to most topologists. By a cellular simple homotopy equivalence, I mean a map of CW-complexes as in 3. It seems to me, that it is somewhat folklore knowledge, that the second setting is a generalization of the prior in the following sense:
Let $K$,$K'$ be abstract simplicial complexes and $f:|K| \to |K'|$ a map between their realizations. Then $f$ is a simplicial simple homotopy equivalence, if and only if it is a cellular simple homotopy equivalence, with respect to the obvious $CW$-structures on $|K|$ and $|K'|$.
Is this even true? I imagined this would be a simple consequence of the simplicial approximation theorem, but couldn't figure out an easy proof. I also skimmed most of the papers from the time period I could find on the matter, but didn't really get a satisfying answer.
If yes, I would be really thankful for a reference, or a sketch of a proof.