Date: Jan 13, 2013 5:06 AM
Subject: Problem with understanding proofs of Sperner's lemma in higher dimensions
Firstly, I'll say that I mean Sperner's lemma on triangulation -- the one that's used to prove Brouwer's theorem.
There are an enormous number of proofs of this on the net and I have the same concerns about all of them for higher dimensions.
The proofs don't develop any simplicial theory beyond defining an n-simplex as the complex hull of n + 1 points in general position.
The proofs and result then become clear for dimensions which can be visualised (such as 1 to 3).
However, for higher dimensions I'm unconvinced because various weird things could arguably happen (but don't happen) that would interfere with the proof.
For example, perhaps you could have 3 n-simplices interesecting in a face which is an n-1 simplex. Of course, this type of thing can't happen. But the way in which I see that this "can't happen" is by picturing the situation in 2 or 3 dimensions and then extending the picture to higher dimensions. This isn't rigorous.
Does anyone know of an account which respects such details? Or, can someone recommend some notes on simplicial theory that would allay such concerns?