Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

Let denote the -dimensional simplex. Let be a random -dimensional subcomplex of obtained by starting with the full -dimensional skeleton of and then adding each -simplex independently with probability . We compute an explicit constant , with , , and as , so that for such a random simplicial complex either collapses to a -dimensional subcomplex or it contains , the boundary of a -dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant such that for any and a fixed field , asymptotically almost surely .