Homological Domination in Large Random Simplicial Complexes

2015/03/11 by A. Costa, Michael Färber, Costa, A. +1 · 2 citations
Computer Science · Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1503.03253

openalex publication_date 2015/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we state the homological domination principle for random multi-parameter simplicial complexes, claiming that the Betti number in one specific dimension (which is explicitly determined by the probability multi-parameter) significantly dominates the Betti numbers in all other dimensions. We also state and discuss evidence for two interesting conjectures which would imply a stronger version of the homological domination principle, namely that generically homology of a random simplicial complex coincides with that of a wedges of k-dimensional spheres. These two conjectures imply that under an additional assumption (specified in the paper) a random simplicial complex collapses to a k-dimensional complex homotopy equivalent to a wedge of spheres of dimension k.

Cited by

• Large random simplicial complexes, I
• Large Random Simplicial Complexes, II; the fundamental groups

Related

• Random Simplicial Complexes, Duality and The Critical Dimension
• Large Random Simplicial Complexes, II; the fundamental groups
• The homology of random simplicial complexes in the multi-parameter upper model
• When does the top homology of a random simplicial complex vanish?
• Limit theorems for Betti numbers of random simplicial complexes