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Homological connectivity of random k-dimensional complexes

R. Meshulam·N. Wallach
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Abstract

Let Delta_{n-1} denote the (n-1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of Delta_{n-1} obtained by starting with the full (k-1)-dimensional skeleton of Delta_{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. Let R and k \geq 1 be fixed. It is shown that p=(k \log n)/n is a sharp threshold for the vanishing of H_{k-1}(Y;R).

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R. Meshulam, N. Wallach. 2006-09-27. Homological connectivity of random k-dimensional complexes. https://arxiv.org/abs/math/0609773

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