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Vanishing of Cohomology Groups of Random Simplicial Complexes (Keynote Speakers)

Authors Oliver Cooley, Nicola Del Giudice, Mihyun Kang, Philipp Sprüssel

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Author Details

Oliver Cooley
  • Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Nicola Del Giudice
  • Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Mihyun Kang
  • Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Philipp Sprüssel
  • Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria

Funding

Supported by Austrian Science Fund (FWF): P27290 and W1230 II

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Oliver Cooley, Nicola Del Giudice, Mihyun Kang, and Philipp Sprüssel. Vanishing of Cohomology Groups of Random Simplicial Complexes (Keynote Speakers). In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.AofA.2018.7

Abstract

We consider k-dimensional random simplicial complexes that are generated from the binomial random (k+1)-uniform hypergraph by taking the downward-closure, where k >= 2. For each 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 monotone, 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. Furthermore, we 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 in [Linial and Meshulam, Combinatorica, 2006], a result which has only been known for dimension two [Kahle and Pittel, Random Structures Algorithms, 2016].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Hypergraphs
Keywords
  • Random hypergraphs
  • random simplicial complexes
  • sharp threshold
  • hitting time
  • connectedness

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References

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