Definable Algebraic Topology

ongoing

This project addresses fundamental issues in the development of algebraic topology, coarse geometry, and other areas of mathematics, related to the problem of doing algebra when the structures under considerations also have a topology. A number of other approaches have been proposed recently, showing the current importance of these issues for the mathematical community. The approach followed in this project is unique, in harnessing powerful tools from mathematical logic, and especially descriptive set theory. The fundamental idea is to enrich an algebraic object with additional information provided by a Polish cover, which is an explicit presentation of the given object as a suitable quotient of a structure endowed with a compatible Polish topology. The goal of this project is to show that fundamental invariants from homological algebra, algebraic topology, operator algebras, and coarse geometry, such as Ext, Cech cohomology, KK-theory, and coarse K-homology, can be seen as functors to the category of groups with a Polish cover. Furthermore, doing so provides invariants that are finer, richer, and more rigid than the purely algebraic ones. These invariants will allow us to tackle classifications problems for topological spaces, coarse spaces, C*-algebras, and maps, that had been so far out of reach. Furthermore, we will use these invariants to calibrate the complexity of such classification problems from the perspective of Borel complexity theory. In turn, this will enable us to isolate complexity-theoretic consequences of the Universal Coefficient Theorem for C*-algebras and of the coarse Baum-Connes Conjecture for coarse spaces, and to construct examples of strong failure of such results. Ultimately, the completion of this project will lead to the development of entirely new fields of research at the interface between logic and other areas of mathematics (algebraic topology, coarse geometry, operator algebras).