A Geometric Approach to Combinatorial Fixed-Point Theorems

TL;DR

This paper introduces a geometric framework unifying various combinatorial fixed-point theorems, leading to new theorems, generalizations, and direct proofs that deepen understanding of topological phenomena.

Contribution

It presents a unified geometric approach to fixed-point theorems, resulting in new theorems, broader generalizations, and direct proofs without relying on topological fixed-point theorems.

Findings

New Tucker-like and Sperner-like fixed-point theorems with exponential label sets

Generalization of Fan's parity proof of Tucker's Lemma

Direct geometric proofs of Sperner-like lemmas from Tucker's lemma

Abstract

We develop a geometric framework that unifies several different combinatorial fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing them to be different geometric manifestations of the same topological phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like fixed-point theorems involving an exponential-sized label set; (2) a generalization of Fan's parity proof of Tucker's Lemma to a much broader class of label sets; and (3) direct proofs of several Sperner-like lemmas from Tucker's lemma via explicit geometric embeddings, without the need for topological fixed-point theorems. Our work naturally suggests several interesting open questions for future research.