Homological connectivity of random k‐dimensional complexes

Let Δ n −1 denote the ( n − 1)‐dimensional simplex. Let Y be a random k ‐dimensional subcomplex of Δ n −1 obtained by starting with the full ( k − 1)‐dimensional skeleton of Δ n −1 and then adding each k ‐simplex independently with probability p . Let H k −1 ( Y ; R ) denote the ( k − 1)‐dimensional reduced homology group of Y with coefficients in a finite abelian group R . It is shown that for any fixed R and k ≥ 1 and for any function ω( n ) that tends to infinity © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009