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.
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.