Is the category of PL manifolds equivalent to the category of topological manifolds in dimension 2,3? Balsam and Kirillov write on page 10 in their paper Turaev Viro invariants as an extended TQFT:

"Note that in dimensions 2 and 3, the category of PL (piecewise linear) manifolds is equivalent to the category of topological manifolds."

This seems like a rather non-trivial result. Why does it hold? Sketching a proof or giving a reference would be very much appreciated.
 A: Here is how this equivalence statement should be understood:

*

*Every topological manifold of dimension $\le 3$ admits a PL structure (equivalently, admits a triangulation: In this range of dimensions there is no difference).


*Such a PL structure is unique in the following sense:
If $M, N$ are PL manifolds of dimensions $\le 3$ then every homeomorphism $f: M\to N$ is isotopic to a PL homeomorphism. In terms of triangulations: The triangulations can be subdivided so that there exists an isomorphism of the subdivided triangulations isotopic to $f$.
The same holds for manifolds with boundary. Furthermore, the same existence/uniqueness holds for locally flat  (equivalently, tame) submanifolds. For instance, if $M$ is a triangulated 3-dimensional manifold and $S\subset M$ is a properly embedded tame surface, then $S$ is properly isotopic to a simplicial subsurface in $M$.
Proofs are rather nontrivial (even the fact that every topological surface admits a triangulation).
Moise, Edwin E., Geometric topology in dimensions 2 and 3, Graduate Texts in Mathematics. 47. New York - Heidelberg - Berlin: Springer-Verlag. X, 262 p. DM 45.00; $ 19.80 (1977). ZBL0349.57001.
Moise, Edwin E., Affine structures in 3-manifolds. V: The triangulation theorem and Hauptvermutung, Ann. Math. (2) 56, 96-114 (1952). ZBL0048.17102.
