Torsion of every finite order in the homology of graph braid groups
Abstract
We determine the torsion subgroup of Hm−1(𝔹mKm+1,m+r−1;ℤ) for m≥2 and r≥0: top homology with arbitrary coefficients is the kernel of an unsigned subset-inclusion matrix, and its integral diagonal form determines all primary summands. Every finite order occurs, with explicit representatives. Generalized theta classes span an embedded copy of the cokernel of the inclusion matrix, containing all torsion; for r≥ m they generate the torsion, each of order lcm(1,…,m). For every prime power q and m≥ q, the graph Km+1,m+q−1 is minimal in the minor order for order-q torsion in Hm−1(𝔹m). In particular, odd torsion first appears in H2(𝔹3K4,5)≅ℤ155⊕(ℤ/2)4⊕ℤ/3, and no proper minor of K4,5 has odd torsion in H2(𝔹3). For arbitrary part sizes, we give a multiplicity-free decomposition of Hm(𝔹mKa,b;ℚ) under vertex permutations and prove that Hm−1(𝔹mKa,b;ℤ) has no p-primary torsion when a,b≥2m−1 and p≥ m is an odd prime. The explicit order-q class retains its order under every enlargement of the second part of the graph, while for m=q=p an odd prime it is killed by a specified enlargement of the first part.
Related
- Mapping class groups of simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology
- A characterization of finite modular 2-groups via magnitude homology
- On a finite group in which every maximal subgroup of order divisible by p containing the normalizer of some Sylow subgroup is nilpotent
- On a finite group in which every non-nilpotent maximal invariant subgroup of order divisible by p is normal
- Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble
BibTeX
@misc{an2026torsion,
title = {{Torsion of every finite order in the homology of graph braid groups}},
author = {Byung Hee An},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04305},
eprint = {2610.04305},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04305},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Torsion of every finite order in the homology of graph braid groups
AU - Byung Hee An
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04305
UR - https://arxiv.org/abs/2610.04305
AB - We determine the torsion subgroup of H_m−1(𝔹_mK_m+1,m+r−1;ℤ) for m≥2 and r≥0: top homology with arbitrary coefficients is the kernel of an unsigned subset-inclusion matrix, and its integral diagonal form determines all primary summands. Every finite order occurs, with explicit representatives. Generalized theta classes span an embedded copy of the cokernel of the inclusion matrix, containing all torsion; for r≥ m they generate the torsion, each of order lcm(1,…,m). For every prime power q and m≥ q, the graph K_m+1,m+q−1 is minimal in the minor order for order-q torsion in H_m−1(𝔹_m). In particular, odd torsion first appears in H_2(𝔹_3K_4,5)≅ℤ^155⊕(ℤ/2)^4⊕ℤ/3, and no proper minor of K_4,5 has odd torsion in H_2(𝔹_3). For arbitrary part sizes, we give a multiplicity-free decomposition of H_m(𝔹_mK_a,b;ℚ) under vertex permutations and prove that H_m−1(𝔹_mK_a,b;ℤ) has no p-primary torsion when a,b≥2m−1 and p≥ m is an odd prime. The explicit order-q class retains its order under every enlargement of the second part of the graph, while for m=q=p an odd prime it is killed by a specified enlargement of the first part.
ER -