Is somebody conscious of an elementary proof of the no-retraction theorem that makes use of the Sperner lemma or one other parity argument?
Two-dimensional case can be ample.
I solely know two proofs that use Sperner AND uniform continuity, however they don’t seem to be completely clear to me.
Furthermore I’m wondering if no-retraction may very well be proved from Sperner with no uniform continuity argument? I as soon as noticed a thread the place somebody stated Sperner Lemma and No-retraction are literally equal. However it appears to me this equivalence shouldn’t be apparent, and it appears the proofs all the time revert to some highly effective extra ideas, like uniform convergence.
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