Definable Algebraic Topology

Introduction

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 consideration also have a topology. A number of other approaches have been proposed recently, showing the current importance of these issues for the mathematical community.

Unique Approach

The approach followed in this project is unique, in harnessing powerful tools from mathematical logic, and especially descriptive set theory.

Fundamental Idea

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.

Project Goals

The goal of this project is to show that fundamental invariants from:

1. Homological algebra
2. Algebraic topology
3. Operator algebras
4. 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.

Applications of Invariants

These invariants will allow us to tackle classification problems for:

• Topological spaces
• Coarse spaces
• C*-algebras
• Maps

that had been so far out of reach.

Complexity Calibration

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
• The coarse Baum-Connes Conjecture for coarse spaces

and to construct examples of strong failure of such results.

Conclusion

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).