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In my previous post, thanks to @John Palmieri, I can show that if $\sigma$ is a simplex then I can triangulate $\sigma\times[0,1]$. Using those triangulation, if $\Delta$ is a simplicial complex then I tried to triangulate $\Delta\times[0,1]$. I first thought this just a simple generalization but I found that there is a compatibility issue. I wonder if there some particular way to triangulate $\Delta\times[0,1]$ using the triangulation of $\sigma\times[0,1]$. Could you help?

one potato two potato
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  • As long as $\Delta$ is a finite simplicial complex with vertices ${v_0, v_1, ...}$, it can be viewed as a set of faces of the simplex $\sigma=(v_0, v_1, ...)$. Then a triangulation of $\sigma \times [0,1]$ will induce on on $\Delta \times [0,1]$. – John Palmieri Sep 09 '21 at 05:11
  • @JohnPalmieri Well by the definition of simplex, ${v_0,v_1,...}$ should be an affinely independent set which is not true in general. – one potato two potato Sep 09 '21 at 05:24
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    It is true up to homeomorphism. – John Palmieri Sep 09 '21 at 06:01
  • @JohnPalmieri Could you be more specific? You mean if I embed the vertex set $V(\Delta)$ to $\Bbb R^n$ for some large $n$, then we can make $V(\Delta)$ an affinely independent set? – one potato two potato Sep 09 '21 at 06:53
  • Right. In particular, embed the vertex set as the vertices of a single simplex, like the standard $n$-simplex in $\mathbb{R}^{n+1}$ if there are a total of $n+1$ vertices. – John Palmieri Sep 09 '21 at 16:07

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You can look at a simplicial complex in at least two ways: (a) combinatorially, a simplicial complex consists of a set $V$ of vertices and a collection of subsets of $V$ (the simplices), closed under taking subsets (their faces). If you have such a thing, you can form its (b) geometric realization, for example by gluing together topological simplices along the faces. Or (as long as it's a finite simplicial complex) embed the vertex set $V$ as the vertices of the standard $n$-simplex, if $V$ has $n+1$ elements, and then insert the faces of that simplex as given by whichever subsets of $V$ are present in the simplicial complex. A topological simplicial complex is anything which is homeomorphic to such a geometric realization.

So you can choose a convenient geometric realization, like embedding the whole simplicial complex $K$ as a set of faces of the standard $n$-simplex $\sigma$. Then a triangulation of $\sigma \times I$ will induce a triangulation of $K \times I$.

John Palmieri
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