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(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).

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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, Annals of Mathematics Studies. 54. Princeton, N.J.: Princeton University Press. XI, 112 p. (1966). ZBL0161.20201.

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, Proc. Am. Math. Soc. 18, 701-704 (1967). ZBL0149.41301.

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  • $\begingroup$ @Thank you this is very nice! It seems that we do not need to assume that the two PL structures are this same (Corollary 1 of the Armstrong paper), or am I missing something? I wonder if the general case (i.e. a non PL triangulation) can then be reduced to the PL. $\endgroup$ – user101010 Sep 27 '19 at 11:13
  • $\begingroup$ In the first paragraph of his article, Armstrong defines a triangulation to be a PL-homeomorphism from a realization of a simplicial complex to a PL-manifold, homeomorphism compatible with the PL-structure of the manifold. $\endgroup$ – Ivan Izmestiev Sep 27 '19 at 11:26

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