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 $2^{\lfloor k/2\rfloor }$ for every $k\ge 1$, and construct divided-power cycles attaining this bound. For $k\ge 2$, the homology lattice is not isomorphic to the standard integral hook lattice as an ${\mathfrak{S}}_{2k+1}$-module. For circulant complexes, we determine the integral dihedral homology modules of $\mathrm{Ind}(C_{3s}^{s-1})$, $\mathrm{Ind}(C_{3s+1}^{s-1})$ and $\mathrm{Ind}(C_{3s+2}^{s-1})$, for $s\ge 3$, $s\ge 2$ and $s\ge 3$, 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.

arXiv comment73 pages. The same manuscript and its computational supplement were published as Zenodo v2.0 on 15 October 2026: https://doi.org/10.5281/zenodo.23383591