# Extending a triangulation of the boundary of $M \times I$

(Sorry for what is probably a rather foundational PL-topology question.) By a triangulation of a manifold $$M$$, I mean a homeomorphism with the geometric realization of a simplicial complex, $$h: |K| \to M$$. I am wondering if I am given two triangulations $$h_0 : |K_0| \to M \times \{0\}$$ and $$h_1 : |K_1| \to M \times \{1\}$$, can I find a triangulation $$h : |K| \to M \times I$$ of $$M \times [0,1]$$ that extends $$h_0$$ and $$h_1$$?

By this I mean that $$K_0$$ and $$K_1$$ both include into $$K$$ and composing the geometric realization of either of these inclusions with $$h$$ commutes with $$h_0$$ and $$h_1$$.

I am happy to assume $$M$$ is compact, orientable, and smooth. I don't particularly care about the triangulations being PL (i.e. I don't care if the links of vertices are spheres).

I think, one has to assume that the triangulations are smooth (i.e. restrictions of $$h_i$$ to every simplex are smooth). Then the answer is yes, this is a special case of a theorem by Munkres: a $$C^r$$-triangulation of the boundary of a manifold extends to a $$C^r$$-triangulation of the manifold, see Theorem 10.6 in