拓扑仲裁器
Topological arbiters
摘要：
本文首先研究了拓扑仲裁子，这是一个起源于Poincare-Lefschetz对偶的概念。给定一个n维流形W，拓扑仲裁者将值0或1与W的余维零子流形相关联，服从自然拓扑和对偶公理。例如，在$RP^2$上有一个唯一的仲裁器，它报告基本1周期的位置。相比之下，我们证明了4维空间中存在无数个拓扑仲裁者集合。家庭的仲裁，不诱导同源性，也存在于更高的层面。四维结果背后的技术成分是广义链接切片问题的次要障碍。对于3-球面中的经典链接，该构造依赖于4维中幂零嵌入障碍的存在，特别是由Milnor群反映出来。在更高的维度，新的仲裁者产生使用非平凡平方稳定同伦理论。“拓扑仲裁者”的概念源自渗透和四维手术。这不是本文的目的，以推进这两个主题，而是为了研究这个概念本身。然而，在附录中，我们给出了渗透的应用，以及目前对仲裁者和手术之间关系的理解。附录还介绍了多仲裁器的更一般的概念。性质和应用进行了讨论，包括非同调多仲裁器的建设。
英文摘要：
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.
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