(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).



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


*Munkres, J. R.*, [**Elementary differential topology. Lectures given at Massachusetts Institute of Technology, Fall, 1961. Revised ed**](http://dx.doi.org/10.1515/9781400882656), Annals of Mathematics Studies. 54. Princeton, N.J.: Princeton University Press. XI, 112 p. (1966). [ZBL0161.20201](https://zbmath.org/?q=an:0161.20201).


In the PL category (where again one should assume that the triangulations define the same PL structure) this is a corollary in the article


*Armstrong, M. A.*, [**Extending triangulations**](http://dx.doi.org/10.2307/2035442), Proc. Am. Math. Soc. 18, 701-704 (1967). [ZBL0149.41301](https://zbmath.org/?q=an:0149.41301).


