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Topological arbiters
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 , 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.Comment: v3: A minor reorganization of the pape
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