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

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 \(B^{lf}_{\mathrm{fin}}(\infty)\subseteq B^{lf}(\infty)\).