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Question 1: Is this the mainstream definition of a PL-manifold?

Definition. A PL-manifold is a manifold with an atlas $(\varphi_i)_{i\in I}$ in which all transition maps $\varphi_j\circ\varphi_i^{-1}$ are piecewise linear. A map between open sets is piecewise linear, if there are (not necessarily finite, but locally finite) triangulations of the domain and range such that the map is linear (affine) restricted to each simpelx.

Question 2: It seems to be well-known fact that PL-manifolds admit a triangulation. But I can't find a proof in the literature or even a reference. All papers talk about much more advanced stuff like triangulating smooth manifolds or giving counterexamples for the reverse (triangulated manifolds which are not PL). I am interested in seeing a proof for this "simple" fact that PL-manifolds are triangulable. I would also like to see a constructive proof, if possible.

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    $\begingroup$ Usually PL maps are defined in terms of them being locally PL, via charts, just like how smoothness is defined locally via charts. The map being PL with respect to a triangulation is then a consequence of the triangulability theorem. For Q2, one way to get this is to follow Whitehead's argument about triangulability of smooth manifolds and see what it gives you in the PL setting where things become easier. Perhaps there is an individual reference for this result, but I suspect most people just think of Whitehead's argument. $\endgroup$
    – Ryan Budney
    Commented Dec 18, 2023 at 19:26
  • $\begingroup$ In Rourke and Sanderson a PL manifold is assumed to be a simplicial complex (with PL charts to Euclidean space). $\endgroup$
    – Ian Agol
    Commented Dec 19, 2023 at 19:54
  • $\begingroup$ I definitely suggest the book of Rourke and Sanderson for everything concerning PL. $\endgroup$
    – Bruno Martelli
    Commented Dec 20, 2023 at 9:43

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