Vanishing of cohomology groups of random simplicial complexes
Oliver Cooley, Nicola Del Giudice, Mihyun Kang, Philipp Sprüssel · Random Structures and Algorithms · 2019
We consider k ‐dimensional random simplicial complexes generated from the binomial random ( k + 1)‐uniform hypergraph by taking the downward‐closure. For 1 ≤ j ≤ k − 1, we determine when all cohomology groups with coefficients in from dimension one up to j vanish and the zero‐th cohomology group is isomorphic to . This property is not deterministically monotone for this model, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the j ‐th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced by Linial and Meshulam, previously only known for dimension two.