Topology of Random 2-Complexes

D. Cohen, A. Costa, M. Farber, T. Kappeler

Research output: Contribution to journalArticlepeer-review


We study the Linial-Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for p≪n-1 a random 2-complex Y collapses simplicially to a graph and, in particular, the fundamental group π1(Y) is free and H2(Y)=0, asymptotically almost surely. Our other main result gives a precise threshold for collapsibility of a random 2-complex to a graph in a prescribed number of steps. We also prove that, if the probability parameter p satisfies p≫n-1/2+ε, where ε>0, then an arbitrary finite two-dimensional simplicial complex admits a topological embedding into a random 2-complex, with probability tending to one as n→∞. We also establish several related results; for example, we show that for p<c/n with c<3 the fundamental group of a random 2-complex contains a non-abelian free subgroup. Our method is based on exploiting explicit thresholds (established in the paper) for the existence of simplicial embeddings and immersions of 2-complexes into a random 2-complex.

Original languageEnglish
Pages (from-to)117-149
Number of pages33
JournalDiscrete and Computational Geometry
Issue number1
StatePublished - Jan 2012
Externally publishedYes


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