Definition of piecewise linear triangulation?

Question

I find it's definition in wikipedia. But it is not self-contained (PL-sphere is not defined there) and does not list reference.

I am wondering is there any reliable reference giving the definition of PL-triangulation? (Some properties of it would be even better.)

Answer 1

Here's a simple definition of a PL triangulation of a manifold, by induction on dimension.

For the basis step, any triangulation of a 1-manifold is PL. In particular, every triangulated circle is PL.

Proceeding by induction, assuming that PL triangulations have been defined in dimension $n-1$, a triangulation of an $n$ dimensional manifold $M$ is PL if and only if for every vertex $v$ of $M$, the link of $v$ in $M$ is a PL triangulation of an $n-1$ dimensional sphere (the link of $v$ being the union of all simplices $\sigma$ such that $v \not\in \sigma$ and such that there exists a simplex $\tau$ such that $v \in \sigma \subset \tau$).