Topology of random 2-complexes

Abstract

We study the Linial--Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for $p\ll n^{-1}$ a random 2-complex $Y$ collapses simplicially to a graph and, in particular, the fundamental group ${\pi }_1(Y)$ is free and $H_2(Y)=0$, a.a.s. We also prove that, if the probability parameter $p$ satisfies $p\gg n^{-1/2+\epsilon }$, where $\epsilon >0$, then an arbitrary finite two-dimensional simplicial complex admits a topological embedding into a random 2-complex, with probability tending to one as $n\to \infty$. 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 nonabelian free subgroup. Our method is based on exploiting explicit thresholds (established in the paper) for the existence of simplicial embedding and immersions of 2-complexes into a random 2-complex.

References in corpus (2)

• The fundamental group of random 2-complexes
• Novikov-Betti numbers and the fundamental group

Cited by in corpus (1)

• Collapsibility and vanishing of top homology in random simplicial complexes