Connectivity of ample, conic and random simplicial complexes

TL;DR

This paper investigates the connectivity properties of simplicial complexes with certain conic conditions, extending previous results and showing that high-conic complexes are highly connected with high probability.

Contribution

It establishes new bounds on the connectivity of conic complexes, generalizing prior work and answering open questions about their topological properties.

Findings

8-conic complexes are 2-connected

(2n+1)-conic complexes may not be n-connected

Probability of n-connectedness tends to 1 as vertices increase

Abstract

A simplicial complex is $r$-conic if every subcomplex of at most $r$ vertices is contained in the star of a vertex. A $4$-conic complex is simply connected. We prove that an $8$-conic complex is $2$-connected. In general a $(2n+1)$-conic complex need not be $n$-connected but a $6^n$-conic complex is $n$-connected. This extends results by Even-Zohar, Farber and Mead on ample complexes and answers two questions raised in their paper. Our results together with theirs imply that the probability of a complex being $n$-connected tends to $1$ as the number of vertices tends to $\infty$. Our model here is the medial regime.