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+ , where > 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.
We study fundamental groups of clique complexes associated to random Erdős–Rényi graphs Γ. We establish thresholds for a number of properties of fundamental groups of these complexes XΓ. In particular, if p=nα, then we show that
4pt1emgdim(π1(XΓ))=cd(π1(XΓ))=1ifα<−12,gdim(π1(XΓ))=cd(π1(XΓ))=2if−12<α<−1130,gdim(π1(XΓ))=cd(π1(XΓ))=∞if−1130<α<−13,
asymptotically almost surely (a.a.s.), where gdim and cd denote the geometric dimension and cohomological dimension correspondingly. It is known that the fundamental group π1(XΓ) is trivial for α>−13. We prove that for −1130<α<−13 the fundamental group π1(XΓ) has 2‐torsion but has no m‐torsion for any given prime m⩾3. We also prove that aspherical subcomplexes of the random clique complex XΓ satisfy the Whitehead conjecture, that is, all their subcomplexes are also aspherical, a.a.s.
In this paper we introduce and develop the multi-parameter model of random simplicial complexes with randomness present in all dimensions. Various geometric and topological properties of such random simplicial complexes are characterised by convex domains in the high-dimensional parameter space (rather than by intervals, as in the usual one-parameter models). We find conditions under which a multi-parameter random simplicial complex is connected and simply connected. Besides, we give an intrinsic characterisation of the multi-parameter probability measure. We analyse links of simplexes and intersections of multi-parameter random simplicial complexes and show that they are also multi-parameter random simplicial complexes.
In this paper we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in [8], [9], [10]. This model includes as special cases the Linial-MeshulamWallach model [19], [20] as well as the clique complexes of random graphs. We characterise the concept of critical dimension in terms of various geometric and topological properties of random simplicial complexes such as their Betti numbers, the fundamental group, the size of minimal cycles and the degrees of simplexes. We mention in the text a few interesting open questions.
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