Definable Algebraic Topology

Project Information

DAT

Grant agreement ID: 101077154

DOI

10.3030/101077154

EC signature date 6 December 2022

Start date 1 January 2023

End date 31 December 2027

Funded under

• European Research Council (ERC)

Total cost

€ 989 395,00

EU contribution

€ 989 395,00

Coordinated by

ALMA MATER STUDIORUM - UNIVERSITA DI BOLOGNA
Italy

Periodic Reporting for period 1 - DAT (Definable Algebraic Topology)

Reporting period: 2023-01-01 to 2025-06-30

Summary of the context and overall objectives of the project

Algebraic topology applies algebraic methods to problems in topology, the study of mathematical structures whose properties are preserved (invariant) under continuous deformation. The ERC-funded DAT project will develop a unique approach to algebraic topology harnessing mathematical logic, specifically descriptive set theory. Algebraic objects will be enriched with additional information provided by a so-called Polish cover, providing invariants that are finer, richer and more rigid than purely algebraic ones. These invariants will enable access to classification problems that were previously out of reach, opening the door to comprehensive study and enhanced understanding of such problems and new areas of research at the interface between logic and other areas of mathematics.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The project so far has isolated the (left heart of the) category of pro-Lie Polish abelian groups as the right category to consider in order to enrich classical algebraic invariants with additional topological information. This category is proved to be well-behaved from the perspective of homological algebra, and its injective and projective objects are completely classified. Furthermore, the left heart of the category of (pro-Lie) Polish abelian groups and all of its thick subcategories have been given an explicit characterization that makes them easy to work with and study. The main achievement of the project so far has been to completely determine the complexity of Ext groups of countable abelian groups in both the torsion and torsion-free cases. In fact, the more general case of modules over a PID and beyond has also been settled. In particular, a strong Dichotomy Theorem has been obtained that produces examples of arbitrarily high complexity. Furthermore, the Solecki subgroups of Ext have been fully characterized in terms of the notion of higher order phantom extensions. Finally, the foundations on the applications to topology of these methods have been laid out by realizing Cech cohomology as the cohomology group of a complex of Polish groups. Via the UCT this provides a correspondence between Solecki subgroups of Ext and higher order phantom maps between spaces, which allows one to apply the results about Ext to precisely compute the latter using the former.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The results obtained so far have wide-reaching implications for several areas of mathematics, due to the fact that a number of algebraic invariants and their complexity can be expressed in terms of the homological invariant Ext for modules by the Universal Coefficients Theorem (UCT). The results concerning Ext, particular will allow one to uncover the phenomenon of phantom objects and high complexity in any instance when mathematical structures can be parametrized by homological invariants for which a suitable UCT holds.

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