Homological Domination in Large Random Simplicial Complexes

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.Comment: 8 pages, 1 figur

A Cheeger-Buser-Type inequality on CW complexes

We extend the definition of boundary expansion to CW complexes and prove a Cheeger-Buser-Type relation between the spectral gap of the Laplacian and the expansion of an orientable CW complex

The threshold for integer homology in random d-complexes

Let Y ~ Y_d(n,p) denote the Bernoulli random d-dimensional simplicial complex. We answer a question of Linial and Meshulam from 2003, showing that the threshold for vanishing of homology H_{d-1}(Y; Z) is less than 80d log n / n. This bound is tight, up to a constant factor.Comment: 12 pages, updated to include an additional torsion group boun