Definable Obstruction Theory

TL;DR

This paper extends classical obstruction theory into the realm of definable algebraic topology, applying descriptive set theory to analyze the complexity of homotopy relations and cohomology groups in Polish spaces.

Contribution

It establishes a definable version of a classical obstruction theory theorem and characterizes the Solecki groups of Cech cohomology for Polish spaces.

Findings

Homotopy relation complexity on continuous maps is characterized.

A definable obstruction theory theorem is proved.

Solecki groups of Cech cohomology are characterized.

Abstract

A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing them as groups with a Polish cover. This allows one to apply techniques from descriptive set theory to the study of cohomology theories. In this paper, we will establish a `definable' version of a classical theorem from obstruction theory, and use this to study the potential complexity of the homotopy relation on the space of continuous maps $C(X, |K|)$, where $X$ is a locally compact Polish space, and K is a locally finite countable simplicial complex. We will also characterize the Solecki Groups of the Cech cohomology of X, which are the canonical chain of subgroups with a Polish cover that are least among those of a given complexity.