# Bringing set theory and algebraic topology together

Lead Research Organisation:
University of Leeds

Department Name: Pure Mathematics

### Abstract

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.

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.

### Planned Impact

The short term impacts of this research will mostly be academic or of a public engagement in science nature. However, as noted on the EPSRC "Geometry and Topology" portfolio webpage, "[r]esearch in geometry and topology is a fundamental cornerstone of all parts of modern mathematics and has underpinning relevance throughout the physical and life sciences, engineering and ICT." The project as a whole may thus be viewed as a "Pathway to Impact" for the field of set theory, promoting possible future economic and societal impacts of the field by way of its influence on algebraic topology.

Academic Impacts are described in more detail in "Academic Beneficiaries".

Direct Economic and Societal Impact:

In the short term, the main economic and societal impacts of this research will be focused around public engagement in science, and the associated enhancement to quality of life and cultural enrichment. In particular, one of the central Objectives of the research is to prove an impossibility theorem about the ability of homology to distinguish homotopy types. Such impossibility theorems, with Goedel's celebrated Incompleteness Theorems as canonical examples, are particularly potent for catching the public imagination. Moreover, homotopy equivalence lends itself to beautiful illuminating diagrams, and connects to another relatively recent news-worthy mathematical event - the resolution of the Poincare Conjecture by Grigori Perelman, for which he was offered and contentiously turned down the Fields medal, the "Nobel Prize of Mathematics". For these reasons, this aspect of the project, when successful, will be ideal for promotion. With assistance from the University of Bristol Press Office, the PI will ensure such promotion takes place, in the popular science press (such as New Scientist magazine for example) and more specifically mathematical general interest publications (such as the American Mathematical Monthly, according to some sources the world's most widely read mathematics journal, with readership including for example many school teachers).

The PI also has a history of outreach and engagement at the schools level, which he intends to continue, inspiring the next generation to follow the STEM pathway. In particular, the fact that the PI researches "different sizes of infinity" is something that can be conveyed to even a young audience, hinting at the often mind-blowing topics that are studied in modern mathematics.

Long term:

It is of course impossible to predict the longer-term impact of such fundamental research. Set theory, and in particular strong axioms about "very large infinities", may seem particularly removed from applications, but this is arguably naive. For example, there are natural statements about simple, finitary algebraic systems known as LD-systems that have only ever been proven under some of the strongest known axioms, and these LD-systems are important for fuller understanding of the mathematics of strings braiding together. It is even theoretically possible that the existence of solutions to the Navier-Stokes equation, which encapsulates our basic understanding of fluid flow, may depend on such strong axioms. Future non-academic applications are eminently possible, but we won't know what they are until we get there.

Academic Impacts are described in more detail in "Academic Beneficiaries".

Direct Economic and Societal Impact:

In the short term, the main economic and societal impacts of this research will be focused around public engagement in science, and the associated enhancement to quality of life and cultural enrichment. In particular, one of the central Objectives of the research is to prove an impossibility theorem about the ability of homology to distinguish homotopy types. Such impossibility theorems, with Goedel's celebrated Incompleteness Theorems as canonical examples, are particularly potent for catching the public imagination. Moreover, homotopy equivalence lends itself to beautiful illuminating diagrams, and connects to another relatively recent news-worthy mathematical event - the resolution of the Poincare Conjecture by Grigori Perelman, for which he was offered and contentiously turned down the Fields medal, the "Nobel Prize of Mathematics". For these reasons, this aspect of the project, when successful, will be ideal for promotion. With assistance from the University of Bristol Press Office, the PI will ensure such promotion takes place, in the popular science press (such as New Scientist magazine for example) and more specifically mathematical general interest publications (such as the American Mathematical Monthly, according to some sources the world's most widely read mathematics journal, with readership including for example many school teachers).

The PI also has a history of outreach and engagement at the schools level, which he intends to continue, inspiring the next generation to follow the STEM pathway. In particular, the fact that the PI researches "different sizes of infinity" is something that can be conveyed to even a young audience, hinting at the often mind-blowing topics that are studied in modern mathematics.

Long term:

It is of course impossible to predict the longer-term impact of such fundamental research. Set theory, and in particular strong axioms about "very large infinities", may seem particularly removed from applications, but this is arguably naive. For example, there are natural statements about simple, finitary algebraic systems known as LD-systems that have only ever been proven under some of the strongest known axioms, and these LD-systems are important for fuller understanding of the mathematics of strings braiding together. It is even theoretically possible that the existence of solutions to the Navier-Stokes equation, which encapsulates our basic understanding of fluid flow, may depend on such strong axioms. Future non-academic applications are eminently possible, but we won't know what they are until we get there.

### Organisations

## People |
## ORCID iD |

Andrew Brooke-Taylor (Principal Investigator / Fellow) |

### Publications

BROOKE-TAYLOR A
(2019)

*THE QUANDARY OF QUANDLES: A BOREL COMPLETE KNOT INVARIANT*in Journal of the Australian Mathematical Society
Brooke-Taylor A
(2017)

*Invariant universality for quandles and fields*
Brooke-Taylor A
(2017)

*Inhabitants of interesting subsets of the Bousfield lattice*
Brooke-Taylor A
(2016)

*Accessible images revisited*in Proceedings of the American Mathematical Society
Brooke-Taylor A
(2018)

*Inhabitants of interesting subsets of the Bousfield lattice*in Journal of Pure and Applied Algebra
Brooke-Taylor A
(2017)

*Cardinal characteristics at ? in a small u ( ? ) model*in Annals of Pure and Applied Logic
Brooke-Taylor A
(2017)

*Products of CW complexes*
Miller S
(2020)

*Invariant universality for quandles and fields*in Fundamenta MathematicaeDescription | The title of the grant is "Bringing set theory and algebraic topology together". The research funded by this grant produced some notable results bringing these areas together. These include the following. - The preferred spaces of study in algebraic topology are called "CW complexes". However, they can behave badly when you try to take the product space of two of them - the product of CW complexes need not be a CW complex. I have found a complete characterisation of when the product of two CW complexes is a CW complex; it depends on how many constituent cells the CW complexes have, compared to an important cardinal from set theory. - Knots can be classified by algebraic structures called "quandles", but there was dissatisfaction in the knot theory community with this means of classification, as quandles seemed to be difficult to tell apart. In collaboration with Sheila Miller of New York City College of Technology, we showed rigorously that this is the case: the problem of telling quandles apart is (in a precise, set-theoretic sense) much harder than the problem of telling knots apart. - The Bousfield lattice B is an important structure in algebraic topology, giving an ordering on homology theories in some sense in terms of their power. There are substructures of B known as DL and cBA which can help to understand the structure of B, but there was no concrete example known of a homology theory in DL but not cBA. In joint work with Benedikt Loewe (University of Amsterdam and University of Hamburg) and Birgit Richter (University of Hamburg), we have shown that a standard homology theory provides such an example. Our approach to the proof was focused on lattice-theoretic properties of B. - A very successful area of set theory for its applications to other areas of mathematics is descriptive set theory, and in particular the study of Borel reducibility. With Filippo Calderoni (University of Turin) I have laid the groundwork to bring a category-theoretic perspective into this area. This will allow it to approach areas such as algebraic topology, where the morphisms of the category-theoretic perspective are crucial. The work has also opened up other new directions for research. In joint work with Jiri Rosicky (Masaryk University Brno), we found that the same tools that have previously been used in applying set theory to algebraic topology could also be used to obtain results in abstract model theory. We have published our first results in this line, and plan to expand research in this direction. |

Exploitation Route | The goal of this research was to bring together two disparate areas of mathematics. The point of this is that the cross-fertilisation one gets can be a boon to both areas. |

Sectors | Other |

Description | There is a clear causal thread from the support this fellowship grant provided for my career to the current status of Leeds as a major centre for Set Theory on the world stage. This might sound like purely academic impact, but one must remember that the UK Higher Education system is heavily subsidised by the fees paid by international students, and it is the straightforward academic excellence of UK universities that drives many foreign students to enrol in them. As such under the current financial model, maintaining the excellence of the pure, blue-skies research produced is crucial to supporting UK universities, and all the cultural, societal, economic and public services benefits they provide. To draw out this thread: it was through this fellowship that I was appointed to a position in Leeds, and moreover through the research that it supported to establish myself on the world stage as one of only a handful of leading figures globally in applications of set theory to category theory and algebraic topology. This position of recognition continues to this day; the most recent example of evidence of this being an invitation to give a tutorial at an upcoming workshop on Set Theory and Algebra in Virginia, USA (https://stationarysetsandalgebra2024.wordpress.com/). As an example of the academic impact flowing immediately from the fellowship, my work with Rosicky is an important piece of the puzzle pinning down the mathematical strength of important model-theoretic ideas in a general context, and as such guiding the development of the theory in this general context. Model theory has had great success throwing light on areas such as number theory (in turn used throughout cryptography, for applications such as secure online banking), but there are many subfields of mathematics that don't fit the standard model-theoretic framework. A more general theory has arisen, with our result guiding its further development particularly in terms of what are suitable mathematical assumptions to build the theory on. This more general theory covers for example Hilbert spaces, central to quantum field theory, which allows us to understand the world at the nano-scale, for possible applications ranging from fusion power generation to materials design. But again, this research is at the very foundations, building our knowledge to feed into such applications in the much longer term. |

Description | Fields Institute Thematic Program Travel Grant |

Amount | $4,000 (CAD) |

Organisation | Fields Institute for Research in Mathematical Sciences |

Sector | Charity/Non Profit |

Country | Canada |

Start | 04/2023 |

End | 06/2023 |

Description | Accessible categories workshop |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other audiences |

Results and Impact | A workshop was held to bring together researchers from very different areas of mathematics all connected through the notion of an accessible category, to encourage new connections and collaboration. |

Year(s) Of Engagement Activity | 2018 |

URL | http://www1.maths.leeds.ac.uk/~pmtadb/AccessibleCategories2018/ |