Binary Voltage Covers of $K(10,3)$: Cohomology, Symmetry Orbits, and a Locally $K(7,3)$ Graph

Abstract

We construct a connected graph on 240 vertices in which every open neighborhood is isomorphic to $K(7,3)$. The graph arises as a binary voltage cover of $K(10,3)$. More generally, the gauge classes of local-neighborhood-preserving binary voltage covers over the fixed labeled base $K(10,3)$ are naturally identified with $H^1(M_3(10);\mathbb{F}_2)$, a vector space of dimension 42. Quotienting by the natural $S_{10}$ action gives 1,245,395 orbits, including 1,245,394 nonzero orbits, each consisting of connected covers. The cohomology class $[\alpha]$ of the displayed 240-vertex graph has $S_{10}$-orbit size 126 and $\operatorname{Stab}_{S_{10}}([\alpha])\cong S_5\wr C_2$. Thus fixed-base covers are classified cohomologically, while allowing base relabeling gives the stated $S_{10}$-orbit set.