The lemmas of Alexander and Sperner

Abstract

Alexander's lemma is a version of Sperner's lemma published by Alexander two years earlier than Sperner's paper. The present paper is devoted to a modern but elementary exposition of lemmas of Alexander and Sperner and their main topological applications: Brouwer's theorems about the topological invariance of dimension and of domains (here we follow Lebesgue ideas in the form given to them by Sperner), Brouwer's fixed-point theorem, and Alexander's theorem about the topological invariance of homology groups. Along the way we relate the Knaster-Kuratowski-Mazurkiewich argument with the notion of simplicial approximations, provide a cohomological interpretation of Sperner's lemma and of its combinatorial proof, and explain how classical proofs of Sperner's and Alexander's lemma lead to path-following algorithms. The exposition does not assume any knowledge of algebraic topology.