Bringing set theory and algebraic topology together

集合论和代数拓扑学是数学的两个主要领域，直到最近才有很少的互动。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在各个领域的独特专业知识。有希望解决代数拓扑学中重要的公开问题，将新概念引入集合论研究的主流，并将这两个领域紧密结合在一起，发展出全新的研究思路。将追求四条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的Borel可约性分析来回答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。这个方向的结果肯定会很有趣：低复杂性将是令人惊讶的，与代数拓扑学中的直觉背道而驰。另一方面，高度的复杂性似乎有深刻的后果，可能暗示了用于区分同伦不等价空间的代数拓扑学的标准工具的根本不足。集合论在局部化中的应用：Bousfield类是代数拓扑学中的重要概念，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Hovey的猜想始终是错误的，建立在最近暗示两种Bousfield类之间的区别的工作之上。一个相关的问题是，是否可以存在一类适当的上同调Bousfield类；PI的目的是使用类似的技巧来证明这实际上是可能的。代数拓扑陈述的大基数强度：已知所有上同调理论的Bousfield局部化的存在源于集合论中被称为大基数公理的强公理。相反，证明大型基数公理的力量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。这一领域的其他声明也仍然被精确地夸大了它们的优势，其中一个特别有趣的例子是弱小的伏本卡原则。支持集合论：PI的一个大型基数不可毁坏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，而不必担心打破大型基数假设。对于较弱的大型基本假设，类似的结果将在已知技术的基础上得到验证，这将是研究方案的宝贵工具。

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.