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Fundamental groups and descriptive set theory

Fanxin Wu

TL;DR

The paper investigates the Borel complexity of the homotopy relation on loop spaces in path-connected Polish spaces, using descriptive set theory to classify when analytic equivalence relations arise as homotopy relations. It introduces the F(E) construction, which translates an analytic equivalence relation E into a loop-homotopy relation, and proves a Becker-style realization showing that every analytic E can be realized as a homotopy relation for some space. An obstruction is established showing that not all analytic relations are realizable, while concrete realizations are provided for canonical relations such as E_0, E_∞, E_1, and the universal relation; the study is extended to complex spaces like the harmonic archipelago and Griffiths space, where HA and GS yield Borel relations above E_1. The work culminates with open questions on bireducibility between HA and GS and discussion of how complexity is preserved or transformed through the F(E) construction.

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.

Fundamental groups and descriptive set theory

TL;DR

The paper investigates the Borel complexity of the homotopy relation on loop spaces in path-connected Polish spaces, using descriptive set theory to classify when analytic equivalence relations arise as homotopy relations. It introduces the F(E) construction, which translates an analytic equivalence relation E into a loop-homotopy relation, and proves a Becker-style realization showing that every analytic E can be realized as a homotopy relation for some space. An obstruction is established showing that not all analytic relations are realizable, while concrete realizations are provided for canonical relations such as E_0, E_∞, E_1, and the universal relation; the study is extended to complex spaces like the harmonic archipelago and Griffiths space, where HA and GS yield Borel relations above E_1. The work culminates with open questions on bireducibility between HA and GS and discussion of how complexity is preserved or transformed through the F(E) construction.

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.

Paper Structure

This paper contains 5 sections, 17 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Homotopy of loops
  3. Obstruction
  4. Realization
  5. Examples

Key Result

Theorem 1

Suppose $X$ is a compact Polish space that is path-connected and locally path-connected. If $X$ is semi-locally simply-connected (slsc), then $\pi(X)$ is finitely generated; if $X$ is not slsc, then $\Omega X$ has a perfect set of non-homotopic loops.

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (35)

  • Theorem : Shelah
  • Definition 1.1
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 25 more