Year: 2026
Type: preprint
Abstract: We determine the torsion subgroup of $H_{m-1}(\mathbb{B}_mK_{m+1,m+r-1};\mathbb{Z})$ for $m\ge2$ and $r\ge0$: 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 $r\ge m$ they generate the torsion, each of order $\operatorname{lcm}(1,\ldots,m)$. For every prime power $q$ and $m\ge q$, the graph $K_{m+1,m+q-1}$ is minimal in the minor order for order-$q$ torsion in $H_{m-1}(\mathbb{B}_m)$. In particular, odd torsion first appears in $H_2(\mathbb{B}_3K_{4,5})\cong\mathbb{Z}^{155}\oplus(\mathbb{Z}/2)^4\oplus\mathbb{Z}/3$, and no proper minor of $K_{4,5}$ has odd torsion in $H_2(\mathbb{B}_3)$. For arbitrary part sizes, we give a multiplicity-free decomposition of $H_m(\mathbb{B}_mK_{a,b};\mathbb{Q})$ under vertex permutations and prove that $H_{m-1}(\mathbb{B}_mK_{a,b};\mathbb{Z})$ has no $p$-primary torsion when $a,b\ge2m-1$ and $p\ge m$ 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.
Source: arXiv (Cornell University)
Author: An, Byung Hee
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220720269