2009/06/29 by Nikolai V. Ivanov, Ivanov, Nikolai V.
Mathematics · Computer Science · #Mathematics and Applications #History and Theory of Mathematics #Constraint Satisfaction and Optimization
paper · pdf · doi:10.48550/arxiv.0906.5193
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.