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Topology of random 2-complexes


Cohen, D; Costa, A; Farber, M; Kappeler, T (2012). Topology of random 2-complexes. Discrete & Computational Geometry, 47(1):117-149.

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.

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.

Citations

18 citations in Web of Science®
18 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2012
Deposited On:07 Feb 2013 07:50
Last Modified:05 Apr 2016 16:27
Publisher:Springer New York LLC
ISSN:0179-5376
Publisher DOI:https://doi.org/10.1007/s00454-011-9378-0
Related URLs:http://arxiv.org/abs/1006.4229

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