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I was working on an example of an abstract simplicial complex and its geometric realization and I had a question. I think the question makes more sense when explained through the example. Some definitions before the example:

An abstract n-simplex, $\Delta_n$, is the set of all non-empty subsets of {0,...,1}.

The geometric standard n-simplex, $|\Delta_n|$ = {$x =(x_0,...,x_n) \in \mathbb{R}^{n+1} | \sum_{i=0}^{n} x_i = 1$}.

A geometric realization of $\Delta_n$ is $|\Delta_n|$ as a subspace of $\mathbb{R}^{n+1}$.

Now for the example, let $V$ be the vertex set {$A,B,C,D$} with 1-simplices {$AB,AC,AD,BC$} and 2-simplices {$ABC$}.

This simplex can be embedded in $\mathbb{R}^2$. Here I mean embedded in the sense that it can be realized as a set of points in $\mathbb{R}^2$. But when {$BCD$} is added, the simplex can no longer be represented in $\mathbb{R}^2$. See here, The abstract simplex and its geometric realization

My question is, given a $\Delta_n$ that can be embedded in $\mathbb{R}^m$ (where $m$ is the minimum dimension possible), what simplices can be added while maintaining the property that $\Delta_n$ can be embedded in $\mathbb{R}^m$. Does it have to do with the size of the simplex being added or does it deal with the number of simplices being added? Is there even any connection between the minimum possible embedding dimension and the type/number of simplices?

Side note: I brought this up to my professor and he said it might be related to when graphs are planar.

Ethan
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1 Answers1

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Thanks for your curiosity and nice pictures. I don't have much time but I'm working on similar stuff right now so I'll drop something.

I think your notation is a bit off. Your $V$ is not a simplex. Note that $\Delta_n$ is intented to be a fixed object that only depends on $n$.

Does it have to do with the size of the simplex being added or does it deal with the number of simplices being added? Is there even any connection between the minimum possible embedding dimension and the type/number of simplices?

It has to do as well with the size (I assume by this you mean dimension) of the simplices you add, the number and their configuration. A simplicial complex with $n$ vertices can always be embedded in $\mathbb{R}^n$. Menger proved in 1928 that every $d$-dimensional simplicial complex can be embedded in $\mathbb{R}^{2d+1}$.

Indeed, embedding twodimensional complexes in $\mathbb{R}^3$ is related to planar graphs, because in general we have the following. Let $\Sigma$ be a simplicial complex and $\mathbf{v}$ a vertex of $\Sigma$. Consider the subcomplex $\Sigma'$ of $\Sigma$ consisting of all those cells of $\Sigma$ that donot contain $\mathbf{v}$. If $\Sigma'$ can be embedded in $\mathbb{R}^d$, then $\Sigma$ can be embedded in $\mathbb{R}^{d+1}$.

Substantial work has been done on understanding which complexes can be embedded in which dimensions of Euclidean space. But it's not easy. You might want to have a look at the following freely available papers:

Maurice Dekker
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  • Where can I find Menger’s result? I do not know if it is a known fact, but I only find the result with the condition over d+2 points. – andrés Mar 14 '25 at 22:19
  • I could only find it buried in his German book "Dimensionstheorie" iirc – Maurice Dekker Mar 16 '25 at 06:35
  • Do you know if this embedding be done isometrically? At least for 1-dimensional simplicial complex. – andrés Mar 16 '25 at 20:03
  • I don't think you can do that isometrically, even for 1-dimensional complexes. Consider the one-dimensional skeleton of a regular tetrahedron. In R^3 there is only one point that has equal distance to all four vertices, namely the tetrahedron's centre. But in R^4 there are infinitely many such points. – Maurice Dekker Mar 18 '25 at 16:21