Simplicial complexes from finite projective planes and colored configurations

Abstract

In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as $T_1\bigsqcup T_2$, such that each $T_i$ represents the lines of a copy of the Fano plane $PG(2,{\mathbb{F}}_2)$. We generalize this observation by constructing, for each prime power $q$, a simplicial complex $X_q$ with $q^2+q+1$ vertices and $2(q^2+q+1)$ facets consisting of two copies of $PG(2,{\mathbb{F}}_q)$. Our construction works for any colored $k$-configuration, defined as a $k$-configuration whose associated bipartite graph $G$ is connected and has a $k$-edge coloring $\chi :E(G)\to [k]$, such that for all $v\in V(G)$, $a,b,c\in [k]$, following edges of colors $a,b,c,a,b,c$ from $v$ brings us back to $v$. We give one-to-one correspondences between (1) Sidon sets of order 2 and size $k+1$ in groups with order $n$, (2) linear codes with radius 1 and index $n$ in the lattice $A_k$, and (3) colored $(k+1)$-configurations with $n$ points and $n$ lines. (The correspondence between (1) and (2) is known.) As a result, we suggest possible topological obstructions to the existence of Sidon sets, and in particular, planar difference sets.