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

TL;DR

This paper investigates the topological fundamental groups of infinite configuration spaces and their quotients, revealing their non-discrete, complete topological group structures and their relation to inverse limits of finite braid groups.

Contribution

It characterizes the fundamental groups of infinite configuration spaces as inverse limits of finite braid groups and constructs compatible ultrametrics, advancing understanding of their topological and algebraic structures.

Findings

Identifies the fundamental groups as inverse limits of finite pure braid groups.

Shows these groups are non-discrete, complete topological groups.

Constructs a compatible ultrametric topology for these groups.

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 $\Aut(\N)$: \[ H^{lf}(\infty):=\pi_1^{\mathrm{top}}(Conf^{lf}_\infty(\C),\widetilde{\N}), \qquad B^{lf}(\infty):=\pi_1^{\mathrm{top}}\!\bigl(Conf^{lf}_\infty(\C)\!/\!/\Aut(\N),[e_0,\widetilde{\N}]\bigr). \] 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 Ra\u{\i}kov completion of this subgroup. Moreover, the direct limit of finite braid groups embeds into \(B^{lf}(\infty)\) and is dense in the finitary subgroup \(B^{lf}_{\mathrm{fin}}(\infty)\subseteq B^{lf}(\infty)\).