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/K035703/1
• 负责人: Andrew Brooke-Taylor
• 金额: $ 53.67万
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FK035703%2F1
• 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，直到最近它们的相互作用还很少。这种情况最近开始发生变化，但由于缺乏具有适当双重专业知识的研究人员，进展缓慢。该项目旨在发展这种新生的联系，充分利用 PI 在各领域独特的专业知识。解决代数拓扑中重大的开放性问题、将新概念引入集合论研究主流以及发展紧密结合这两个领域的全新研究路线都有前景。将追求四个紧密交织的研究线索：1。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性集合论中的 Borel 可还原性分析来回答 C* 代数理论中的问题。本项目将使用这些技术进行类似的程序来研究同伦等价，即代数拓扑中的基本关系。这个方向的结果一定会很有趣：低复杂度会令人惊讶，这与代数拓扑中的直觉背道而驰。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。2．应用于局部化的集合论：布斯菲尔德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，Hovey 猜想每个上同调 Bousfield 类也是同调 Bousfield 类。这仍然是一个重要的悬而未决的问题，但在这个项目中，PI 打算证明霍维的猜想始终是错误的，以最近暗示两种 Bousfield 类之间的区别的工作为基础。一个相关的问题是是否存在上同调 Bousfield 类的真类？ PI 旨在表明，使用类似的技术实际上这是可能的。3。代数拓扑陈述的大基数强度：已知所有上同调理论的 Bousfield 定域性的存在都遵循集合论中称为大基数公理的强公理。相反，证明大基数公理的强度对于上同调定位是必要的，这将是非常有趣的，甚至可能会改变该领域的观点。该领域的其他陈述也仍然对其优势进行了精确的衡量，弱沃本卡原理就是一个特别有趣的例子。4。支持集合论：PI 的大基数不可破坏性定理已被证明与该领域的研究相关，允许相当自由地使用强制的核心技术，而不必担心破坏大基数假设。较弱的大基数假设的类似结果将通过建立在已知技术的基础上得到证明，这将是该研究计划的宝贵工具。

项目成果

Cichon's Diagram for uncountable cardinals

不可数基数的 Cichon 图

• DOI: 10.48550/arxiv.1611.08140
• 发表时间: 2016
• 作者: Brendle J

Canonical Ramsey Theory on Polish Spaces (Encyclopedia of Mathematics and its Applications 202) By Vladimir Kanovei, Marcin Sabok and Jindrich Zapletal

波兰空间的规范拉姆齐理论（数学及其应用百科全书 202） 作者：Vladimir Kanovei、Marcin Sabok 和 Jindrich Zapletal

• DOI: 10.1112/blms/bdv016
• 发表时间: 2015
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Brooke-Taylor A

The quandary of quandles: The Borel completeness of a knot invariant

Quandles 的困境：结不变量的 Borel 完备性

• DOI: 10.48550/arxiv.1602.03209
• 发表时间: 2016
• 作者: Brooke-Taylor A

An Analogy between Cardinal Characteristics and Highness Properties of Oracles

神谕的基本特征与殿下属性的类比

• DOI: 10.1142/9789814678001_0001
• 发表时间: 2015
• 作者: Brendle J

Complexity of a knot invariant

结不变量的复杂性

• 发表时间: 2016
• 作者: A. Brooke-Taylor