Existence of subdivision of PL manifold triangulation which is combinatorial manifold Suppose $X$ is a PL manifold with triangulation $\psi:|\Delta| \to X$. Does there exist a subdivision of $\psi$ which is a combinatorial triangulation?

Assuming $\psi$ is not combinatorial, no, never, because subdivision preserves the homeomorphism type of the link of a vertex. (In particular, if it's not a sphere, you're out of luck.) This is reasonably straightforward to prove.