Homological Domination in Large Random Simplicial Complexes

TL;DR

This paper introduces the homological domination principle for large random simplicial complexes, showing that Betti numbers in a specific dimension dominate others, and discusses conjectures linking homology to wedges of spheres.

Contribution

It establishes the homological domination principle and explores conjectures connecting homology of random complexes to wedges of spheres, advancing understanding of their topological structure.

Findings

Betti number in one dimension dominates others

Evidence for conjectures linking homology to wedges of spheres

Conditions under which complexes collapse to wedges of spheres

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.