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Topological fundamental groups of locally finite infinite configuration spaces and infinite braids

Jyh-Haur Teh

TL;DR

This work develops the topology on the locally finite infinite configuration spaces in the plane by endowing the associated topological braid groups $H^{lf}( ty)$ and $B^{lf}( ty)$ with the quotient topology from based loop spaces. It proves non-discreteness and complete metrizability for these groups, and constructs a canonical locally finite inverse-limit model built from finite pure braid groups, yielding a dense embedding of the direct limit $P_ ty$ and identifying $H^{lf}( ty)$ as the Raïkov completion of this subgroup. The paper also analyzes the full braid group, showing completeness and a clopen decomposition into finitary and non-finitary parts, with a precise closure $\,ar{B}_ ty=\\pi_*^{-1}(\\mathrm{Sym}_f(\mathbb{N}))$. Furthermore, it proves Polishness for the pure infinite braid group and delineates which completions yield Polishness for the various subgroups, ultimately highlighting a novel braid-at-infinity topology distinct from classical finite braids. These results establish a robust topological framework for infinite braids and locally finite configurations, with implications for completeness, density, and completion phenomena in infinite-dimensional braid-like objects.

Abstract

We study the topological fundamental groups of the locally finite infinite ordered configuration space \(Conf^{lf}_\infty(\C)\) in the plane and the homotopy quotient of $Conf^{lf}_\infty$ by the canonical action of the infinite permutation group $\Aut(\N)$: \[ H^{lf}(\infty):=π_1^{\mathrm{top}}(Conf^{lf}_\infty(\C),\widetilde{\N}), \qquad B^{lf}(\infty):=π_1^{\mathrm{top}}\!\bigl(Conf^{lf}_\infty(\C)\!/\!/\Aut(\N),[e_0,\widetilde{\N}]\bigr). \] We prove that \(H^{lf}(\infty)\) and \(B^{lf}(\infty)\) are non-discrete and complete topological groups. A main structural theorem identifies \(H^{lf}(\infty)\) with a canonical locally finite inverse-limit model built from finite pure braid groups, and we construct a complete left-invariant ultrametric compatible with the quotient topology from the loop space of $\Conf$. The direct limit of finite pure braid groups admits a dense embedding into \(H^{lf}(\infty)\), and we show that \(H^{lf}(\infty)\) is the Raĭkov completion of this subgroup. Moreover, the direct limit of finite braid groups embeds into \(B^{lf}(\infty)\) and is dense in the finitary subgroup \(B^{lf}_{\mathrm{fin}}(\infty)\subseteq B^{lf}(\infty)\).

Topological fundamental groups of locally finite infinite configuration spaces and infinite braids

TL;DR

This work develops the topology on the locally finite infinite configuration spaces in the plane by endowing the associated topological braid groups and with the quotient topology from based loop spaces. It proves non-discreteness and complete metrizability for these groups, and constructs a canonical locally finite inverse-limit model built from finite pure braid groups, yielding a dense embedding of the direct limit and identifying as the Raïkov completion of this subgroup. The paper also analyzes the full braid group, showing completeness and a clopen decomposition into finitary and non-finitary parts, with a precise closure . Furthermore, it proves Polishness for the pure infinite braid group and delineates which completions yield Polishness for the various subgroups, ultimately highlighting a novel braid-at-infinity topology distinct from classical finite braids. These results establish a robust topological framework for infinite braids and locally finite configurations, with implications for completeness, density, and completion phenomena in infinite-dimensional braid-like objects.

Abstract

We study the topological fundamental groups of the locally finite infinite ordered configuration space \(Conf^{lf}_\infty(\C)\) in the plane and the homotopy quotient of by the canonical action of the infinite permutation group : \[ H^{lf}(\infty):=π_1^{\mathrm{top}}(Conf^{lf}_\infty(\C),\widetilde{\N}), \qquad B^{lf}(\infty):=π_1^{\mathrm{top}}\!\bigl(Conf^{lf}_\infty(\C)\!/\!/\Aut(\N),[e_0,\widetilde{\N}]\bigr). \] We prove that \(H^{lf}(\infty)\) and \(B^{lf}(\infty)\) are non-discrete and complete topological groups. A main structural theorem identifies \(H^{lf}(\infty)\) with a canonical locally finite inverse-limit model built from finite pure braid groups, and we construct a complete left-invariant ultrametric compatible with the quotient topology from the loop space of . The direct limit of finite pure braid groups admits a dense embedding into \(H^{lf}(\infty)\), and we show that \(H^{lf}(\infty)\) is the Raĭkov completion of this subgroup. Moreover, the direct limit of finite braid groups embeds into \(B^{lf}(\infty)\) and is dense in the finitary subgroup \(B^{lf}_{\mathrm{fin}}(\infty)\subseteq B^{lf}(\infty)\).
Paper Structure (38 sections, 60 theorems, 263 equations)

This paper contains 38 sections, 60 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Topological fundamental groups and completeness
  3. Topological fundamental groups and completely metrizable spaces
  4. The locally finite pure braid group $H^{lf}(\infty)$
  5. The vague metric and the sum metric
  6. A uniform finiteness lemma
  7. Discreteness criteria
  8. Non-semilocal simple connectivity of $(Conf^{lf}_{\infty}(\mathbb{C}),d_{\sum})$ and non-discreteness of $H^{lf}(\infty)$
  9. Inverse limits and the canonical map $\Theta$
  10. Locally finite inverse limit
  11. Continuity and bijectivity of the canonical map $\Theta$
  12. The continuity and surjectivity of $\Theta$
  13. The injectivity of $\Theta$
  14. A compatible complete left-invariant metric on $H^{lf}(\infty)$
  15. Moving strands
  16. ...and 23 more sections

Key Result

Theorem 1

The sequence eq:lf-braid-seq-intro is exact as a sequence of topological groups: the inclusion $H^{lf}(\infty)\hookrightarrow B^{lf}(\infty)$ is a topological embedding onto a closed normal subgroup, and the quotient topology on $B^{lf}(\infty)/H^{lf}(\infty)$ identifies it with the discrete group $

Theorems & Definitions (132)

  • Theorem : Topological exact sequence
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Quasitopological group structure
  • proof
  • Definition 2.4: Complete metrizability
  • Lemma 2.5
  • proof
  • Definition 2.6: The vague metric $d_{\mathcal{V}}$
  • Lemma 2.7: Uniform metric on loop spaces
  • ...and 122 more