A crystallographic quotient of the framed braid group and associated flat manifolds
Abstract
We study the quotient of the framed braid group FBn by the commutator subgroup of the framed pure braid group FPn. We prove that FBn/[FPn,FPn]≅ℤnrtimesBn/[Pn,Pn], and, for n≥3, that this quotient is a crystallographic group of dimension n(n+1)/2 with holonomy group Sn. We investigate its finite-order elements and conjugacy classes, prove that it has no elements of finite even order, and construct Bieberbach subgroups with prescribed finite $2$-group holonomy. Starting from suitable Bieberbach subgroups of Bn/[Pn,Pn], we then construct Bieberbach subgroups of the framed quotient whose associated compact flat manifolds are torus bundles over the corresponding unframed flat manifolds. We describe these bundles through the generalized Calabi construction and determine their orientability, first Betti numbers, and the centers of their fundamental groups. As an application, we study a family with cyclic holonomy of odd order, obtaining a criterion for the existence of flat Kähler structures, and indicate how the same construction applies to cyclic holonomy of $2$-power order.
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BibTeX
@misc{leite2026crystallographic,
title = {{A crystallographic quotient of the framed braid group and associated flat manifolds}},
author = {Ênio Leite and Oscar Ocampo and Paulo Cesar Cerqueira dos Santos Júnior and Ana Carolina Moura Teixeira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33308},
eprint = {2610.33308},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33308},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A crystallographic quotient of the framed braid group and associated flat manifolds
AU - Ênio Leite
AU - Oscar Ocampo
AU - Paulo Cesar Cerqueira dos Santos Júnior
AU - Ana Carolina Moura Teixeira
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33308
UR - https://arxiv.org/abs/2610.33308
AB - We study the quotient of the framed braid group FB_n by the commutator subgroup of the framed pure braid group FP_n. We prove that FB_n/[FP_n,FP_n]≅ℤ^nrtimesB_n/[P_n,P_n], and, for n≥3, that this quotient is a crystallographic group of dimension n(n+1)/2 with holonomy group S_n. We investigate its finite-order elements and conjugacy classes, prove that it has no elements of finite even order, and construct Bieberbach subgroups with prescribed finite $2$-group holonomy. Starting from suitable Bieberbach subgroups of B_n/[P_n,P_n], we then construct Bieberbach subgroups of the framed quotient whose associated compact flat manifolds are torus bundles over the corresponding unframed flat manifolds. We describe these bundles through the generalized Calabi construction and determine their orientability, first Betti numbers, and the centers of their fundamental groups. As an application, we study a family with cyclic holonomy of odd order, obtaining a criterion for the existence of flat Kähler structures, and indicate how the same construction applies to cyclic holonomy of $2$-power order.
ER -