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40 years or so ago, a kid named Jeremy showed me a proof of the two-dimensional Brouwer fixed-point theorem, which used what I have since come to know is called "Sperner's lemma." The two-dimensional proof he showed me was actually stronger than Sperner's lemma. It used a Stokes' Theorem-type argument to show that the complete triangles can be assigned a value +1 or -1 depending on their orientation (1-2-3 vs 3-2-1, viewed from some consistent orientation), and that the total of these values must be 1. Sperner's lemma follows by taking this mod 2.

I would love to have a published reference for this. I'm sure it holds in higher dimensions if you're willing to go to the painful trouble of orienting your simplices.

And Jeremy, if you're out there, I'd like to thank you.

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    $\begingroup$ Connected : Tucker's lemma as in this nice presentation : pretty.structures.free.fr/talks/Meunier.pdf ? $\endgroup$
    – Jean Marie
    Commented Jun 15, 2020 at 21:35
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    $\begingroup$ Are you thinking of the lemma Stronger form of Sperner on page 3 or this PDF? $\endgroup$ Commented Jun 15, 2020 at 21:42
  • $\begingroup$ Thanks, @BrianM.Scott that is what I'm thinking about. The references show a link to planetmath which is dead (but this one seems correct: planetmath.org/spernerslemma), which has the same theorem with no references. Hopefully Michael Henle's book or (less likely) the rental harmony article may be helpful. $\endgroup$
    – Nat Kuhn
    Commented Jun 16, 2020 at 23:06
  • $\begingroup$ @JeanMarie Thanks, that is indeed a nice presentation, but not what I'm looking for. $\endgroup$
    – Nat Kuhn
    Commented Jun 16, 2020 at 23:07

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