Sperner's Lemma, Brouwer's fixed-point theorem, and cohomology

TL;DR

This paper explores the deep connections between Sperner's lemma, Brouwer's fixed-point theorem, and cohomology, showing how combinatorial proofs relate to algebraic topology concepts.

Contribution

It demonstrates that Sperner's lemma can be viewed as a cochain-level cohomological argument and clarifies the topological reasoning behind Brouwer's theorem.

Findings

Sperner's lemma corresponds to a cochain-level cohomological proof.

The deduction of Brouwer's theorem from Sperner's lemma parallels the no-retraction argument.

The paper provides a self-contained explanation linking combinatorial topology and algebraic topology.

Abstract

The proof of Brouwer's fixed-point theorem based on Sperner's lemma is often presented as an elementary combinatorial alternative to advanced proofs based on algebraic topology. The goal of this note is to show that: (i) 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; (ii) the standard deduction of Brouwer's theorem from Sperner's lemma is similar to the usual deduction of Brouwer's theorem from the no-retraction theorem and is closely related to the notion of a simplicial approximation. In order to make these connections transparent, we included the above mentioned standard arguments, so the note is self-contained modulo some basic ideas of combinatorial topology.