Is there a direct proof that an $(n-1)$-simplex in a subdivision of the standard $n$-simplex is a face of at most two of its $n$-simplices?

Question

There are many books and articles that prove Sperner's Lemma. Almost all that I have looked up happily take the following as obvious.

If $\mathcal{S}$ is a simplicial subdivision of the standard $n$-simplex, then every $(n-1)$-simplex of $\mathcal{S}$ is a face of at most two $n$-simplices of $\mathcal{S}$.

Most of the references don't even state this explicitly. Some of them prove the lemma only under an explicit assumption that the above holds. The only one that goes into a deeper discussion is a Polish book Wstęp do topologii by Engelking and Sieklucki. (I think that "Engelking, Sieklucki Topology: a geometric approach Sigma Series in Pure Mathematics, 4. Heldermann Verlag, Berlin, 1992" might be an English translation but I'm not sure since I have no access to it.) At first, they only prove Sperner's Lemma for iterated barycentric subdivisions for which the property above can be verified directly. Later, they show (using Invariance of Domain) that this property holds for all subdivisions.

My question: is there an elementary proof of the statement above? By "elementary" I mean one that avoids things like Invariance of Domain as well as homotopical and homological arguments.

Answer 1

It depends on what you mean by a "simplicial subdivision". I can think of two different definitions, leading to two different outcomes for your question.

In one definition, a "simplicial subdivision" is obtained from the original decomposition into simplices by repeating some kind of elementary subdivision. The iterated barycentric subdivision is like this, but there are more general constructions. In that case the proof should simply be induction.

In another defnition, a "simplicial subdivsion" simply means another simplicial structure on the same space of which the original skeleta are subcomplexes. In this situation, invariance of domain is your friend where nothing else will help, it seems to me.

The theme here is that the local topology of Euclidean space is subtler than you might think. Try proving that dimension is a topological invariant without homological arguments, for example.