Vanishing of cohomology groups of random simplicial complexes. Issue 2 (23rd April 2019)

• Record Type: Journal Article
• Authors: Cooley, Oliver; Giudice, Nicola Del; Kang, Mihyun; Sprüssel, Philipp
• Is Part Of: Random structures & algorithms. Volume 56:Issue 2(2020)
• Page Start: 461
• Page End: 500
• Publication Date: 2019-04-23
• Subjects: connectedness -- hitting time -- random hypergraphs -- random simplicial complexes -- sharp threshold; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20857
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract : 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 F 2 from dimension one up to j vanish and the zero‐th cohomology group is isomorphic to F 2 . 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.