Celebratio Mathematica — Freedman

The A-B slice problem

Works connected to V. Slava Krushkal

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article M. Freedman and V. Krushkal: “Topological arbiters,” J. Topol. 5 : 1 (February 2010), pp. 226–247. ArXiv 1002.1063

This paper initiates the study of topological arbiters, a concept rooted in Poincaré–Lefschetz duality. Given an \( n \)-dimensional manifold \( W \), a topological arbiter associates a value 0 or 1 to codimension-0 submanifolds of \( W \), subject to natural topological and duality axioms. For example, there is a unique arbiter on \( \mathbb{R}P^2 \), 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 4-dimensional results are secondary obstructions to generalized link-slicing problems. For classical links in \( \mathbb{S}^3 \) 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 non-trivial 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 the 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.