Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

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 = Θ(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. We conjecture this bound to be sharp. In addition we show that there exists a constant γd < cd < d+ 1 such that for any c > cd and a fixed field F, asymptotically almost surely Hd(Y ; F) 6= 0.