Periodic Reporting for period 1 - DAT (Definable Algebraic Topology)
Période du rapport: 2023-01-01 au 2025-06-30
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.
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.
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.