Fundamental groups and descriptive set theory

TL;DR

This paper explores the homotopy of loops in Polish spaces through descriptive set theory, revealing how various analytic equivalence relations can be characterized and examining the properties of the free group over such relations.

Contribution

It introduces a descriptive set-theoretic perspective to the homotopy of loops and analyzes the complexity of associated equivalence relations and free groups.

Findings

Many analytic equivalence relations arise from homotopy of loops.

Certain equivalence relations do not correspond to such homotopies.

The structure of the free group over an equivalence relation is studied.

Abstract

We study the homotopy of loops in a fixed path-connected Polish space from a descriptive set-theoretic viewpoint. We show that many analytic equivalence relations arise this way, and many do not. We also study the "free group" over an equivalence relation.