Vanishing of cohomology groups of random simplicial complexes

O. Cooley, N.D. Giudice, M. Kang, P. Sprüssel*

*Corresponding author for this work

Research output: Contribution to journalArticle

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 F2 from dimension one up to j vanish and the zero‐th cohomology group is isomorphic to F2. 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.
Original languageEnglish
Pages (from-to)461-500
Number of pages40
JournalRandom Structures & Algorithms
Volume56
Issue number2
DOIs
Publication statusPublished - 2020

Keywords

  • connectedness
  • hitting time
  • random hypergraphs
  • random simplicial complexes
  • sharp threshold

ASJC Scopus subject areas

  • Software
  • Applied Mathematics
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design

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