Homological Connectivity Of Random 2-Complexes

Let Δ_n−1 denote the (n − 1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of Δ_n−1 obtained by starting with the full 1-dimensional skeleton of Δ_n−1 and then adding each 2−simplex independently with probability p. Let $H_{1} {\left( {Y;{\Bbb F}_{2} } \right)}$ denote the first homology group of Y with mod 2 coefficients. It is shown that for any function ω(n) that tends to infinity

$$ {\mathop {\lim }\limits_{n \to \infty } }{\kern 1pt} {\kern 1pt} {\text{Prob}}{\left[ {H_{1} {\left( {Y;{\Bbb F}_{2} } \right)} = 0} \right]} = \left\{ {\begin{array}{*{20}c} {{0p = \frac{{2\log n - \omega {\left( n \right)}}} {n}}} \\ {{1p = \frac{{2\log n + \omega {\left( n \right)}}} {n}}} \\ \end{array} } \right. $$