Topology of random 2-complexes
Cohen, D., Costa, A., Farber, Michael and Kappeler, T.. (2012) Topology of random 2-complexes. Discrete & Computational Geometry, Vol.47 (No.1). pp. 117-149. ISSN 0179-5376Full text not available from this repository.
Official URL: http://dx.doi.org/10.1007/s00454-011-9378-0
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.
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Mathematics|
|Journal or Publication Title:||Discrete & Computational Geometry|
|Publisher:||Springer New York LLC|
|Page Range:||pp. 117-149|
|Access rights to Published version:||Restricted or Subscription Access|
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