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Homological connectivity of random 2-complexes. (English) Zbl 1121.55013
Summary: Let \(\Delta_{n-1}\) denote the \((n - 1)\)-dimensional simplex. Let \(Y\) be a random 2-dimensional subcomplex of \(\Delta_{n-1}\) obtained by starting with the full 1-dimensional skeleton of \(\Delta_{n-1}\) and then adding each 2-simplex independently with probability \(p\). Let \( H_1(Y;\mathbb F_2)\) denote the first homology group of \(Y\) with mod 2 coefficients. It is shown that for any function \(\omega(n)\) that tends to infinity \[ \lim_{n\to \infty} \text{Prob}\,[H_1(Y;\mathbb F_2)=0]=\begin{cases} 0 & p=\dfrac{2\log n-\omega(n)}{n}\\ 1 & p=\dfrac{2\log n+\omega(n)}{n}\end{cases}. \]

MSC:
55U10 Simplicial sets and complexes in algebraic topology
05C80 Random graphs (graph-theoretic aspects)
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