Random complexes with free involution

TL;DR

This paper introduces a new model for random simplicial complexes with a free involution, analyzing their typical topology and establishing a probabilistic Borsuk--Ulam theorem, with applications to Erdős–Rényi graphs.

Contribution

The paper develops a novel model for random complexes with a free involution and studies their asymptotic topological properties, including a probabilistic Borsuk--Ulam theorem.

Findings

Complexes often have simply-connected double covers.

Bounds for dimensions where equivariant maps have zeros.

Structural insights into non-adjacent cliques in random graphs.

Abstract

We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group $\mathbb{Z}/2\mathbb{Z}$. We establish results about the typical asymptotic topology of these complexes. As a consequence we give bounds for the dimension $d$ such that $\mathbb{Z}/2\mathbb{Z}$-equivariant maps from the double cover to $\mathbb{R}^d$ have zeros with high probability, thus establishing a random Borsuk--Ulam theorem. We apply this to derive a structural result for pairs of non-adjacent cliques in Erd\H{o}s--R\'{e}nyi random graphs.