Topological arbiters

TL;DR

This paper introduces topological arbiters, functions on submanifolds of manifolds satisfying duality axioms, revealing their existence in various dimensions and exploring their properties and applications in topology and related fields.

Contribution

It defines topological arbiters rooted in duality, proves their existence in dimension 4 and higher, and connects them to link-slicing obstructions and stable homotopy theory.

Findings

Unique arbiter on $RP^2$ reports essential 1-cycle location.

Uncountably many arbiters exist in 4D, not induced by homology.

Higher-dimensional arbiters constructed using stable homotopy theory.

Abstract

This paper initiates the study of topological arbiters, a concept rooted in Poincare-Lefschetz duality. Given an n-dimensional manifold W, a topological arbiter associates a value 0 or 1 to codimension zero submanifolds of W, subject to natural topological and duality axioms. For example, there is a unique arbiter on $RP^2$, which reports the location of the essential 1-cycle. In contrast, we show that there exists an uncountable collection of topological arbiters in dimension 4. Families of arbiters, not induced by homology, are also shown to exist in higher dimensions. The technical ingredients underlying the four dimensional results are secondary obstructions to generalized link-slicing problems. For classical links in the 3-sphere the construction relies on the existence of nilpotent embedding obstructions in dimension 4, reflected in particular by the Milnor group. In higher dimensions novel arbiters are produced using nontrivial squares in stable homotopy theory. The concept of "topological arbiter" derives from percolation and from 4-dimensional surgery. It is not the purpose of this paper to advance either of these subjects, but rather to study the concept for its own sake. However in appendices we give both an application to percolation, and the current understanding of the relationship between arbiters and surgery. An appendix also introduces a more general notion of a multi-arbiter. Properties and applications are discussed, including a construction of non-homological multi-arbiters.