Topological fundamental groups of locally finite infinite configuration spaces and infinite braids
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914313714794496 |
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| author | Teh, Jyh-Haur |
| author_facet | Teh, Jyh-Haur |
| contents | 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 Ra\uıkov completion of this subgroup. Moreover, the direct limit of finite braid groups embeds into \(B^{lf}(\infty)\) and is dense in the finitary subgroup \(B^{lf}_{\mathrm{fin}}(\infty)\subseteq B^{lf}(\infty)\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07362 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological fundamental groups of locally finite infinite configuration spaces and infinite braids Teh, Jyh-Haur Algebraic Topology Mathematical Physics General Topology Group Theory 20F36, 54H11, 57M05 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 Ra\uıkov completion of this subgroup. Moreover, the direct limit of finite braid groups embeds into \(B^{lf}(\infty)\) and is dense in the finitary subgroup \(B^{lf}_{\mathrm{fin}}(\infty)\subseteq B^{lf}(\infty)\). |
| title | Topological fundamental groups of locally finite infinite configuration spaces and infinite braids |
| topic | Algebraic Topology Mathematical Physics General Topology Group Theory 20F36, 54H11, 57M05 |
| url | https://arxiv.org/abs/2602.07362 |