Let $K$ be a PL triangulation of a closed manifold and $f: K\to\mathbb R^k$ be a PL map. Equivalently, $f$ is a map that becomes linear on every simplex after subdividing. Suppose $v$ is a vertex in the **interior** of $f(K)$.

- Is $f^{-1}(v)$ is a locally flat submanifold?
- If so, does it have a PL trivial normal bundle?

I don't think this is answered in any standard PL topology book. All I found is a lemma on p.94 in Kirby-Siebemann's book which says that $f$ is a trivial bundle over the interior of every top-dimensional simplex in $f(K)$.