The Structure of Cycles in Projective Geometry over ${\mathbb{F}}_q$

Abstract

A classical geometric result says that every nonzero cycle of the mod-$2$ incidence map from $d$-subsets to $(d-1)$-subsets of $[n]$ has support at least $d+1$, with equality attained by the boundary of a simplex on $d+1$ vertices. We prove an analogous result for the subspace lattice of ${\mathbb{F}}_q^n$, determining the minimum support size of a nonzero $d$-cycle over a field $K$ of characteristic $p\mid q+1$. Surprisingly, the boundary of a $(d+1)$-space is not always optimal. Shorter cycles occur for $d=1$, and for $d=2$ when $n\ge 4$, and otherwise, the boundary of a $(d+1)$-space is shortest. For $d\ge 4$, we prove a gap-stability result: every cycle with support less than $(2-10/q)$ times the minimum is a multiple of the boundary of a $d+1$ space. We also construct support-controlled cones, yielding a direct geometric analysis of the dimensions in which the homology groups of the subspace incidence complex vanish and explicit lower bounds on its coboundary expansion. The degree-$1$ expansion estimate is an ingredient in the stability theorem.

arXiv comment16 pages. A previous version with major overlaps appeared as 2306.14317