Simplicial complexes from finite projective planes and colored configurations

TL;DR

This paper constructs simplicial complexes from finite projective planes and colored configurations, revealing connections between combinatorial designs, coding theory, and topology, and suggesting obstructions to certain combinatorial structures.

Contribution

It generalizes a known triangulation of the torus using Fano planes to all prime powers and establishes new correspondences linking combinatorial, algebraic, and topological structures.

Findings

Constructed simplicial complexes from projective planes for prime powers.

Established bijections between Sidon sets, linear codes, and colored configurations.

Suggested topological obstructions to the existence of certain combinatorial sets.

Abstract

In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as $T_{1} \sqcup T_{2}$, 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 $\chi \colon E(G) \to [k]$, 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.