Collapsibility and vanishing of top homology in random simplicial complexes

Details Collapsibility and vanishing of top homology in random simplicial complexes Collapsibility and vanishing of top homology in random simplicial complexes

2010-10-07

arxiv.org/abs/1010.1400v2

Mathematics

Combinatorics

Theorem 1.3 in the first version is replaced by the stronger Theorem 1.4

Scientific paper

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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