Integral homology lattices of matching and circulant complexes

Abstract

Bouc's hook cycles span the rational top homology of the matching complex of K_2k+1. We prove that their integral saturation quotient has exponent exactly 2^{⌊ k/2⌋} for every k≥1, and construct divided-power cycles attaining this bound. For k≥2, the homology lattice is not isomorphic to the standard integral hook lattice as an S_2k+1-module. For circulant complexes, we determine the integral dihedral homology modules of Ind(C_3s^s−1), Ind(C_3s+1^s−1) and Ind(C_3s+2^s−1), for s≥3, s≥2 and s≥3, respectively, and classify the least scalar lifts of all primitive equivariant quotients to the rank-one module with trivial rotation and negative reflection. Tate localization relates prime-order gluing to fixed-point homology under a free, single-degree homology hypothesis. Explicit cycle subgroups separate saturation defects from obstructions to equivariant sections. For circulant independence complexes with jumps two and three, we prove an eight-periodic ordinary homotopy classification and a fixed-space obstruction to uniform representation suspensions. For every fixed nonempty bounded positive jump set, a finite-state construction gives a rational critical-face series.

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BibTeX RIS

BibTeX

@misc{wang2026integral,
  title = {{Integral homology lattices of matching and circulant complexes}},
  author = {Qihang Wang and Weiye Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22082},
  eprint = {2610.22082},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22082},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Integral homology lattices of matching and circulant complexes
AU  - Qihang Wang
AU  - Weiye Li
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22082
UR  - https://arxiv.org/abs/2610.22082
AB  - Bouc's hook cycles span the rational top homology of the matching complex of K_2k+1. We prove that their integral saturation quotient has exponent exactly 2^⌊ k/2⌋ for every k≥1, and construct divided-power cycles attaining this bound. For k≥2, the homology lattice is not isomorphic to the standard integral hook lattice as an S_2k+1-module. For circulant complexes, we determine the integral dihedral homology modules of Ind(C_3s^s−1), Ind(C_3s+1^s−1) and Ind(C_3s+2^s−1), for s≥3, s≥2 and s≥3, respectively, and classify the least scalar lifts of all primitive equivariant quotients to the rank-one module with trivial rotation and negative reflection. Tate localization relates prime-order gluing to fixed-point homology under a free, single-degree homology hypothesis. Explicit cycle subgroups separate saturation defects from obstructions to equivariant sections. For circulant independence complexes with jumps two and three, we prove an eight-periodic ordinary homotopy classification and a fixed-space obstruction to uniform representation suspensions. For every fixed nonempty bounded positive jump set, a finite-state construction gives a rational critical-face series.
ER  -

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