Dowker's theorem for higher-order relations

TL;DR

This paper generalizes Dowker's Theorem to multiway relations, establishing homotopy equivalences between complex structures, and provides a detailed analysis of ternary relations with functorial homotopy types.

Contribution

It extends Dowker's Theorem from binary to multiway relations, introducing new homotopy equivalences and analyzing ternary relations in detail.

Findings

Established functorial homotopy equivalences for multiway Dowker complexes

Identified seven homotopy types for ternary relations

Developed a cellular Dowker lemma and a simplexification process

Abstract

Given a relation $R \subseteq I \times J$ between two sets, Dowker's Theorem (1952) states that the homology groups of two associated simplicial complexes, now known as Dowker complexes, are isomorphic. In its modern form, the full result asserts a functorial homotopy equivalence between the two Dowker complexes. What can be said about relations defined on three or more sets? We present a simple generalization to multiway relations of the form $R \subseteq I_1 \times I_2 \times \cdots \times I_m$. The theorem asserts functorial homotopy equivalences between $m$ multiway Dowker complexes and a variant of the rectangle complex of Brun and Salbu from their recent short proof of Dowker's Theorem. Our proof uses Smale's homotopy mapping theorem and factors through a cellular Dowker lemma that expresses the main idea in more general form. To make the geometry more transparent, we work with a class of spaces called prod-complexes then transfer the results to simplicial complexes through a simplexification process. We conclude with a detailed study of ternary relations, identifying seven functorially defined homotopy types and twelve natural transformations between them.