Bringing set theory and algebraic topology together

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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.