Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

@article{Aronshtam2013CollapsibilityAV,
title={Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes},
author={Lior Aronshtam and Nathan Linial and Tomasz Luczak and Roy Meshulam},
journal={Discrete & Computational Geometry},
year={2013},
volume={49},
pages={317-334}
}

Let n−1 denote the (n − 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of n−1 obtained by starting with the full (d − 1)dimensional skeleton of n−1 and then adding each d-simplex independently with probability p = c n . We compute an explicit constant γd , with γ2 2.45, γ3 3.5, and γd = (log d) as d → ∞, so that for c < γd such a random simplicial complex either collapses to a (d − 1)-dimensional subcomplex or it contains ∂ d+1, the boundary of a (d + 1)-dimensional simplex… CONTINUE READING