Fundamental groups and descriptive set theory

2025/12/02 by Wu, Fanxin
#03E15 (Primary) 54H05 #20F34 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2512.02355

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.

Citations

• The Complexity of Arc-Connectedness Relation in the Plane
• The double cone group is isomorphic to the archipelago group
• The fundamental group of the harmonic archipelago

Related

• Aspects of Classical Descriptive Set Theory
• Coverings and fundamental groups: a new approach
• On quasi-small loop groups
• Models of an Abstract Elementary Class as a Generalized Polish Space
• Loops on schemes and the algebraic fundamental group