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Session S27 - Categories and Topology

Friday, July 16, 12:00 ~ 12:30 UTC-3

Connectivity of random simplicial complexes

Jonathan Barmak

A simplicial complex is $r$-conic if every subcomplex of at most $r$ vertices is contained in a cone. We prove that for any $d\ge 0$ 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 $\infty$. Thus, random complexes are $d$-connected with high probability.