Fixed-Point Theorem, and Cohomology

1. INTRODUCTION. The proof of the Brouwer fixed-point Theorem based on Sperner's Lemma [S] is often presented as an elementary combi-natorial alternative to advanced proofs based on algebraic topology. See, for example, Section 6.3 of [P1]. One may ask if this proof is really based on ideas completely different from the ideas of algebraic topology (and, in particular, the ideas of Brouwer's own proof, based on the degree theory, a fragment of algebraic topology developed by him)? After the author discovered [I] that the famous analytic proof of the Brouwer Theorem due to Dunford and Schwartz is nothing else than the usual topological proof in disguise, he started to suspect that the same is true for the proof based on Sperner's Lemma. This suspicion turned out to be correct, and the goal of this note is to uncover the standard topology hidden in this proof. In fact, the two situations are very similar. The Dunford-Schwartz proof can be considered as a cochain-level version of the standard proof based on the de Rham cohomology (cochains of the de Rham theory are differential forms), written in the language of elementary multivariable calculus. Similarly , the combinatorial proof of Sperner's Lemma can be considered as a cochain-level version, written in the combinatorial language, of a standard cohomological argument. This time one needs to use simplicial cohomology and simplicial cochains. It is remarkable that both alternative approaches turn out to be cohomological proofs in disguise, and not the homological ones. It turns out that the standard deduction of the Brouwer Theorem from Sperner's Lemma is also similar to some arguments in algebraic topology. Namely, most proofs of the Brouwer Theorem are based on a construction of a retraction of a disk or a simplex onto its boundary starting from its fixed point free map to itself, combined with the No-Retraction Theorem to the effect that there are no such retractions. It turns out that this construction of a retraction underlies the standard deduction of the Brouwer Theorem from Sperner's Lemma. The No-Retraction Theorem itself can be proved