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Vanishing of cohomology groups of random simplicial complexes
Author(s) -
Cooley Oliver,
Giudice Nicola Del,
Kang Mihyun,
Sprüssel Philipp
Publication year - 2020
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20857
Subject(s) - mathematics , cohomology , čech cohomology , binomial (polynomial) , dimension (graph theory) , equivariant cohomology , combinatorics , simplicial complex , pure mathematics , group (periodic table) , discrete mathematics , de rham cohomology , statistics , chemistry , organic chemistry
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 inF 2from dimension one up to j vanish and the zero‐th cohomology group is isomorphic toF 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.