PreprintCC BY 4.0arXiv
The first homology of square-section braid groups
Abstract
We compute the first homology of the unordered configuration space of n open unit-squares in the w by h rectangle for all positive integers n, w, and h. Letting w and h vary, these square configuration spaces form a bi-filtration of the unordered configuration space of n points in the plane, so their fundamental groups approximate the n-strand braid group. Using a pair of spectral sequences, we prove that these square and point configuration spaces have naturally isomorphic first homology groups if w,h≥ 3 and n≤ wh−3. In the h=2 case, we show that the first homology groups of these square configuration spaces are free of rank n−1 for n≤ 2w−2, whereas they are free of rank w−1 for n=2w−1.
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BibTeX
@misc{pham2026first,
title = {{The first homology of square-section braid groups}},
author = {Minh Pham and Nicholas Wawrykow and Jacob Zweiback},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12633},
eprint = {2610.12633},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12633},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The first homology of square-section braid groups
AU - Minh Pham
AU - Nicholas Wawrykow
AU - Jacob Zweiback
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12633
UR - https://arxiv.org/abs/2610.12633
AB - We compute the first homology of the unordered configuration space of n open unit-squares in the w by h rectangle for all positive integers n, w, and h. Letting w and h vary, these square configuration spaces form a bi-filtration of the unordered configuration space of n points in the plane, so their fundamental groups approximate the n-strand braid group. Using a pair of spectral sequences, we prove that these square and point configuration spaces have naturally isomorphic first homology groups if w,h≥ 3 and n≤ wh−3. In the h=2 case, we show that the first homology groups of these square configuration spaces are free of rank n−1 for n≤ 2w−2, whereas they are free of rank w−1 for n=2w−1.
ER -