$\ell^2$-homology and planar graphs

TL;DR

This paper demonstrates the non-planarity of the complete graph $K_5$ using $ ext{l}^2$-homology techniques, extending prior work that applied similar methods to $K_{3,3}$.

Contribution

It introduces a novel application of $ ext{l}^2$-homology to prove the non-planarity of $K_5$, complementing existing results for $K_{3,3}$.

Findings

Proves $K_5$ is not planar using $ ext{l}^2$-homology.

Extends the method previously applied to $K_{3,3}$.

Provides a new algebraic-topological approach to graph planarity.

Abstract

In his 1930 paper, Kuratowksi categorized planar graphs, proving that a finite graph $\Gamma$ is planar if and only if it does not contain a subgraph that is homeomorphic to $K_5$, the complete graph on 5 vertices, or $K_{3,3}$, the complete bipartite graph on six vertices. In their 2001 paper, Davis and Okun point out that the $K_{3,3}$ graph can be understood as the nerve of a right-angled Coxeter system and prove that this graph is not planar using results from $\ell^2$-homology. In this paper, we employ a similar method proving $K_5$ is not planar.