Homological connectivity of random k-dimensional complexes

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Let Deltaₙ₋₁ denote the (n-1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of Deltaₙ₋₁ obtained by starting with the full (k-1)-dimensional skeleton of Deltaₙ₋₁ and then adding each k-simplex independently with probability p. Let Hₖ₋₁(Y;R) denote the (k-1)-dimensional reduced homology group of Y with coefficients in a finite abelian group R. Let R and k ≥ 1 be fixed. It is shown that p=(k n)/n is a sharp threshold for the vanishing of Hₖ₋₁(Y;R).