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.