I want to say a word about a theorem of Bing, and a consequence I am sure he knew, and I thought everyone knew, so I just assumed it often in talks and papers (I do this often, unfortunately).
I did this several times, until a prominent mathematician said, during a colloquium in Paris, angrily that this could not be true. So maybe it is less folklore than I thought.
Anyway, to the theorem: Bing proved that if a simplicial complex X embeds facewise linearly into a PL manifold M, then M has a triangulation that contains X as a subcomplex. That seems believable, and is not so hard to prove, and contained in R. H. Bing, The geometric topology of 3-manifolds, AMS, 1983.
Now the corollary that people have problems with is this: If X embeds piecewise linearly into a PL manifold M, then M has a triangulation that contains X as a subcomplex.
That is a bit counterintuitive, as the map may be rather wild, and locally not flat. In fact, the theorem requires that we deform the map a little. But it is true nevertheless, and it is what Zuzana Patáková and I proved.
Though again, probably everyone knew anyway 😀
For a cup of coffee more 