Collapsibility and vanishing of top homology in random simplicial complexes
Combinatorics
2011-08-04 v2
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 <d+1 such that for any c>c_d and a fixed field F, asymptotically almost surely H_d(Y;F) \neq 0.
Keywords
Cite
@article{arxiv.1010.1400,
title = {Collapsibility and vanishing of top homology in random simplicial complexes},
author = {L. Aronshtam and N. Linial and T. Luczak and R. Meshulam},
journal= {arXiv preprint arXiv:1010.1400},
year = {2011}
}
Comments
Theorem 1.3 in the first version is replaced by the stronger Theorem 1.4