Bringing set theory and algebraic topology together

将集合论和代数拓扑结合在一起

• 批准号: EP/K035703/2
• 负责人: Andrew Brooke-Taylor
• 金额: $ 25.72万
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2016
• 资助国家: 英国
• 起止时间: 2016 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FK035703%2F2
• 关键词: Bringing; set; theory; algebraic; topology

Set theory and algebraic topology are two major fields of mathematics that until recently have had very little interaction. This has recently started to change, but progress has been slow because of a lack of researchers with appropriate dual expertise. This project aims to develop this nascent connection, making full use of the PI's unique breadth of expertise across the fields. There are prospects for resolving significant open problems in algebraic topology, for introducing new concepts to the mainstream of set-theoretic research, and for the development of whole new lines of inquiry intimately combining the two fields.Four closely interwoven threads of research will be pursued:1. Complexity of homotopy equivalence: One of the most impressive recent applications of set theory has been the use of Borel reducibility analysis from descriptive set theory to answer questions in the theory of C*-algebras. The present project will undertake an analogous programme using these techniques to study homotopy equivalence, the fundamental relation in algebraic topology. Results in this direction are bound to be interesting: low complexity would be surprising, running counter to intuition in algebraic topology. On the other hand, high complexity would seem to have profound ramifications, possibly implying a fundamental inadequacy of the standard tools of algebraic topology for distinguishing homotopy inequivalent spaces.2. Set theory applied to localisation: Bousfield classes are important constructs in algebraic topology, intimately connected with localisation. In a 1995 paper, Hovey conjectured that every cohomological Bousfield class is also a homological Bousfield class. This remains an important open problem, but in this project the PI intends to show that Hovey's conjecture is consistently false, building on recent work hinting at a distinction between the two kinds of Bousfield class. A related question is whether there can be a proper class of cohomological Bousfield classes; the PI aims to show that in fact this is possible, using similar techniques.3. Large cardinal strength of algebraic topology statements: The existence of Bousfield localisations for all cohomology theories is known to follow from strong axioms in set theory known as large cardinal axioms. Showing that conversely, the strength of large cardinal axioms is necessary for cohomological localisation would be extremely interesting and may even change perspectives in the fields. Other statements in the area also remain to have their strengths precisely guaged, with Weak Vopenka's Principle a particularly interesting example.4. Supporting set theory: A large cardinal indestructibility theorem of the PI has already proven relevant to research in this area, allowing fairly free use of the central technique of forcing without fear of breaking large cardinal assumptions. Similar results for weaker large cardinal assumptions, to be proven by building on known techniques, will be an invaluable tool for the research programme.

集合论和代数拓扑是数学的两个主要领域，直到最近才有很少的相互作用。这种情况最近开始改变，但进展缓慢，因为缺乏具有适当双重专业知识的研究人员。该项目旨在发展这种新生的联系，充分利用PI在各个领域独特的专业知识。有前景解决重大开放问题的代数拓扑，为引进新概念的主流集理论研究，并为发展全新的调查线密切结合这两个领域。同伦等价的复杂性：集合论最近最令人印象深刻的应用之一是使用描述集合论的Borel归约分析来回答C*-代数理论中的问题。本项目将进行一个类似的计划，使用这些技术来研究同伦等价，代数拓扑学中的基本关系。在这个方向上的结果一定是有趣的：低复杂性将是令人惊讶的，与代数拓扑学的直觉背道而驰。另一方面，高复杂性似乎会产生深远的影响，可能意味着代数拓扑的标准工具在区分同伦不等价空间方面存在根本性的不足。应用于局部化的集合论：Bousfield类是代数拓扑中的重要构造，与局部化密切相关。在1995年的一篇论文中，霍维证明了每一个上同调的Bousfield类也是一个同调的Bousfield类。这仍然是一个重要的开放问题，但在这个项目中，PI打算证明霍维的猜想是一贯错误的，建立在最近的工作暗示两种Bousfield类之间的区别。一个相关的问题是是否可以有一个适当的类的上同调Bousfield类; PI的目的是表明，事实上这是可能的，使用类似的技术。代数拓扑陈述的大基数强度：已知所有上同调理论的Bousfield局部化的存在性遵循集合论中的强公理，称为大基数公理。相反，证明大基数公理的强度对于上同调局部化是必要的，这将是非常有趣的，甚至可能改变该领域的观点。这一领域的其他陈述也仍然需要精确地衡量它们的优势，弱沃彭卡原则是一个特别有趣的例子。支持集合理论：PI的一个大基数不灭性定理已经被证明与这一领域的研究相关，允许相当自由地使用强迫的核心技术，而不必担心打破大基数假设。对于较弱的大型基本假设，类似的结果将通过利用已知技术来证明，这将成为研究计划的宝贵工具。

项目成果

THE QUANDARY OF QUANDLES: A BOREL COMPLETE KNOT INVARIANT

QUANDLES 的困境：一个 Borel 完全结不变量

• DOI: 10.1017/s1446788719000399
• 发表时间: 2019
• 期刊: Journal of the Australian Mathematical Society
• 影响因子: 0.7
• 作者: BROOKE-TAYLOR A

Invariant universality for quandles and fields

Quantles 和 Field 的不变普遍性

• DOI: 10.4064/fm862-2-2020
• 发表时间: 2020
• 期刊: Fundamenta Mathematicae
• 影响因子: 0.6
• 作者: Miller S

Accessible images revisited

重新审视无障碍图像

• DOI: 10.1090/proc/13190
• 发表时间: 2016
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Brooke-Taylor A

Products of CW complexes

CW 复合体产品

• DOI: 10.48550/arxiv.1710.05296
• 发表时间: 2017
• 作者: Brooke-Taylor A

Inhabitants of interesting subsets of the Bousfield lattice

布斯菲尔德晶格有趣子集的居民

• DOI: 10.1016/j.jpaa.2017.09.012
• 发表时间: 2018
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Brooke-Taylor A