2022/10/20 by Frick, Florian, Newman, Andrew
#05C80 #05E45 #55U10 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2210.11316
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 ℤ/2ℤ. We establish results about the typical asymptotic topology of these complexes. As a consequence we give bounds for the dimension d such that ℤ/2ℤ-equivariant maps from the double cover to ℝd 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 Erdős--Rényi random graphs.