The lemmas of Alexander and Sperner

TL;DR

This paper provides a modern, elementary exposition of Alexander's and Sperner's lemmas, their topological applications, and connections to algorithms, without requiring prior knowledge of algebraic topology.

Contribution

It offers a comprehensive, accessible presentation of key topological lemmas and their applications, linking combinatorial and algebraic topology concepts.

Findings

Unified proof of topological invariance of dimension and domains

Cohomological interpretation of Sperner's lemma

Path-following algorithms derived from classical proofs

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.