eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-18
7:1
7:14
10.4230/LIPIcs.AofA.2018.7
article
Vanishing of Cohomology Groups of Random Simplicial Complexes (Keynote Speakers)
Cooley, Oliver
1
Del Giudice, Nicola
1
Kang, Mihyun
1
Sprüssel, Philipp
1
Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
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].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol110-aofa2018/LIPIcs.AofA.2018.7/LIPIcs.AofA.2018.7.pdf
Random hypergraphs
random simplicial complexes
sharp threshold
hitting time
connectedness