Article thumbnail
Location of Repository

Topology of random 2-complexes

By D. Cohen, A. Costa, Michael Farber and T. Kappeler

Abstract

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 π 1(Y) is free and H 2(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

Topics: QA
Publisher: Springer New York LLC
Year: 2012
OAI identifier: oai:wrap.warwick.ac.uk:43425
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • http://dx.doi.org/10.1007/s004... (external link)
  • Suggested articles


    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.