# Bringing set theory and algebraic topology together

Lead Research Organisation:
University of Bristol

Department Name: 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.

## People |
## ORCID iD |

Andrew Brooke-Taylor (Principal Investigator / Fellow) |

### Publications

A. Brooke-Taylor
(2016)

*Complexity of a knot invariant*
Brendle J
(2016)

*Cichon's Diagram for uncountable cardinals*
Brooke-Taylor A
(2018)

*Inhabitants of interesting subsets of the Bousfield lattice*in Journal of Pure and Applied Algebra
Brooke-Taylor A
(2015)

*Canonical Ramsey Theory on Polish Spaces (Encyclopedia of Mathematics and its Applications 202) By Vladimir Kanovei, Marcin Sabok and Jindrich Zapletal*in Bulletin of the London Mathematical Society
Brooke-Taylor A
(2017)

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

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

*The quandary of quandles: The Borel completeness of a knot invariant*
Brooke-Taylor A
(2016)

*Accessible images revisited*in Proceedings of the American Mathematical Society
BROOKE-TAYLOR A
(2019)

*THE QUANDARY OF QUANDLES: A BOREL COMPLETE KNOT INVARIANT*in Journal of the Australian Mathematical SocietyDescription | This grant was continued at the University of Leeds, so this overlaps significantly with the ResearchFish report for that part of the grant. 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. - 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 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. - 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. The work has also opened up some 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 | I expect there to be much research building on these findings. In the short term the applications will be entirely academic, but the long term is notoriously hard to predict. |

Sectors | Other |

Description | See the "Narrative Impact" for the Leeds component of this fellowship, EP/K035703/2. |

First Year Of Impact | 2018 |

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 | Dobrinen - ultrafilters on measurable kappa |

Organisation | University of Denver |

Country | United States |

Sector | Academic/University |

PI Contribution | Participated in joint research on properties of ultrafilters on measurable cardinals. |

Collaborator Contribution | Dr Natasha Dobrinen participated in joint research on properties of ultrafilters on measurable cardinals. A visit of Dobrinen to Leeds for the research was paid for by Dobrinen's NSF grant. |

Impact | No tangible outputs as yet. |

Start Year | 2016 |

Description | Rosicky |

Organisation | Masaryk University |

Country | Czech Republic |

Sector | Academic/University |

PI Contribution | Contributed to group effort figuring out improvements to an application of set theory to category theory. Paid for travel to Brno, Czech Republic, for the research. Paid for my time in Barcelona during the research programme there at which Rosicky and I continued the collaboration. |

Collaborator Contribution | Contributed to the group effort mentioned above. Paid for my accommodation in Brno and at a research group workshop in Trest. |

Impact | Initial results published as Brooke-Taylor and Rosicky, "Accessible Images Revisited", Proc. Amer. Math. Soc. 145 (2017) no 3, pp 1317-1327. Not multi-disciplinary, but "intra-disciplinary" in the terminology of the EPSRC mathematics priority: set theory applied to category theory to obtain a result with important implications in model theory. |

Start Year | 2014 |

Description | Sheila Miller Borel reducibility |

Organisation | Imperial College London |

Department | Department of Mathematics |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Worked on joint research on Borel reducibility complexity results with connections to knot theory. |

Collaborator Contribution | Dr Sheila Miller worked on joint research on Borel reducibility complexity results with connections to knot theory. |

Impact | A joint paper, published in the Journal of the Australian Mathematical Society: A. Brooke-Taylor and S. Miller, "The quandary of quandles: the Borel completeness of a knot invariant". Not multidisciplinary, but "intradisciplinary". A joint paper, also with Filippo Calderoni, accepted to appear in Fundamenta Mathematicae |

Start Year | 2015 |

Description | The small u(kappa) model collaboration |

Organisation | University of Vienna |

Country | Austria |

Sector | Academic/University |

PI Contribution | Contributed to a group effort calculating and fixing the values of characterics of a certain model; in particular, brought special expertise in the model. Paid for my travel to Vienna for this research to take place. |

Collaborator Contribution | Contributed to the group effort mentioned above, in particular, brought specific expertise in the characteristics considered. Paid for my accommodation in Vienna. |

Impact | Results published as Brooke-Taylor, Fischer, Friedman and Montoya, "Cardinal characteristics at kappa in a small u(kappa) model", Annals of Pure and Applied Logic 168 (2017) no 1, pp37-49. Not multi-disciplinary. |

Start Year | 2014 |