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arXiv · 1806.04566

Vanishing of cohomology groups of random simplicial complexes

Oliver Cooley·Nicola Del Giudice·Mihyun Kang·Philipp Sprüssel
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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 $k\geq 2$. For each $1\leq j \leq k-1$, we determine when all cohomology groups with coefficients in $\mathbb{F}_2$ from dimension one up to $j$ vanish and the zero-th cohomology group is isomorphic to $\mathbb{F}_2$. This property is not deterministically monotone for this model of random complexes, 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 in [Linial and Meshulam, Combinatorica, 2006], a result which was previously only known for dimension two [Kahle and Pittel, Random Structures Algorithms, 2016].

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Oliver Cooley, Nicola Del Giudice, Mihyun Kang, Philipp Sprüssel. 2018-06-12. Vanishing of cohomology groups of random simplicial complexes. https://arxiv.org/abs/1806.04566

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