When does the top homology of a random simplicial complex vanish?

Details When does the top homology of a random simplicial complex vanish? When does the top homology of a random simplicial complex vanish?

2012-03-15

arxiv.org/abs/1203.3312v1

Mathematics

Combinatorics

Scientific paper

Several years ago Linial and Meshulam introduced a model called X_d(n,p) of random n-vertex d-dimensional simplicial complexes. The following question suggests itself very naturally: What is the threshold probability p=p(n) at which the d-dimensional homology of such a random d-complex is, almost surely, nonzero? Here we derive an upper bound on this threshold. Computer experiments that we have conducted suggest that this bound may coincide with the actual threshold, but this remains an open question.

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