Connectivity of random simplicial complexes

BARMAK, JONATHAN ARIEL

Congreso; Mathematical Congress of the Americas in the special session Categories and Topology; 2021

A simplicial complex is r-conic if every subcomplex of at most r vertices is contained in a cone. We prove that for any d≥0 there exists r such that r-conicity implies d-connectivity of the polyhedron. On the other hand, for a fixed r, the probability of a random simplicial complex being r-conic tends to 1 as the number of vertices tends to ∞. Thus, random complexes are d-connected with high probability.