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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)
##### Keywords:
random 2-complexes
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