Haven't read the article. But something about this reminds me of Arnold's topological proof of the unsolvability of the quintic (YouTube form: <a href="https://www.youtube.com/watch?v=BSHv9Elk1MU" rel="nofollow">https://www.youtube.com/watch?v=BSHv9Elk1MU</a> ; PDF: <a href="https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf" rel="nofollow">https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.p...</a>).<p>It seems a lot of impossibility theorems - the type that the ancient Greeks would have understood - can be proven using algebraic topology. Perhaps Sperner's lemma can be seen as an algebraic topology theorem? I don't personally know.
> To show that detM
is non-zero, we can show that its 2-adic valuation is nonzero.<p>I think the last word in that sentence should be "finite"?<p>Also do I understand correctly that "face" means "maximal line segment"? (I see some other comments discussing this and concluding that "face" means "edge", but to me, an "edge" doesn't permit "intermediate" vertices.)
> no face of P, nor any face of one of the Ti, contains vertices of all three colors<p>That should be 'edge', not 'face', no? Otherwise I do not understand what is happening at all with the examples.