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arXiv · 1010.1400

Collapsibility and vanishing of top homology in random simplicial complexes

L. Aronshtam·N. Linial·T. Luczak·R. Meshulam
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Abstract

Let Y be a random d-dimensional subcomplex of the (n-1)-dimensional simplex S obtained by starting with the full (d-1)-dimensional skeleton of S and then adding each d-simplex independently with probability p=c/n. We compute an explicit constant gamma_d=Theta(log d) so that for c < gamma_d such a random simplicial complex either collapses to a (d-1)-dimensional subcomplex or it contains the boundary of a (d+1)-simplex. We conjecture this bound to be sharp. In addition we show that there exists a constant gamma_d< c_d c_d and a fixed field F, asymptotically almost surely H_d(Y;F) \neq 0.

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L. Aronshtam, N. Linial, T. Luczak, R. Meshulam. 2011-08-03. Collapsibility and vanishing of top homology in random simplicial complexes. https://arxiv.org/abs/1010.1400

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