Toric Orbit Spaces Which are Manifolds

Q3 Mathematics Arnold Mathematical Journal Pub Date : 2024-01-03 DOI:10.1007/s40598-023-00242-5

Anton Ayzenberg, Vladimir Gorchakov

引用次数: 0

Abstract

We characterize the actions of compact tori on smooth closed manifolds for which the orbit space is a topological manifold (either closed or with boundary). For closed manifolds, the result was originally proved by Styrt in 2009. We give a new proof for closed manifolds which is also applicable to manifolds with boundary. In our arguments, we use the result of Provan and Billera who characterized matroid complexes which are pseudomanifolds. We study the combinatorial structure of torus actions whose orbit spaces are manifolds. In two appendix sections, we give an overview of two theories related to our work. The first one is the combinatorial theory of Leontief substitution systems from mathematical economics. The second one is the topological Kaluza–Klein model of Dirac’s monopole studied by Atiyah. The aim of these sections is to draw some bridges between disciplines and motivate further studies in toric topology.

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属于流形的环形轨道空间

我们描述了轨道空间为拓扑流形（闭合流形或有边界流形）的光滑闭合流形上紧凑环的作用。对于封闭流形，这一结果最初由斯蒂尔特在 2009 年证明。我们给出了封闭流形的新证明，它也适用于有边界的流形。在论证中，我们使用了普罗旺（Provan）和比勒拉（Billera）的结果，他们描述了作为伪流形的母题复合体的特征。我们研究了轨道空间为流形的环作用的组合结构。在两个附录部分，我们概述了与我们的工作相关的两个理论。第一个是数学经济学中的列昂惕夫替代系统组合理论。第二个是阿蒂亚研究的狄拉克单极的拓扑卡卢扎-克莱因模型。这些章节的目的是在各学科之间架起一些桥梁，并激发对环状拓扑学的进一步研究。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Arnold Mathematical Journal Mathematics-Mathematics (all)

CiteScore

1.50

自引率

0.00%

发文量

期刊介绍： The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.

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