Armindo Costa

Topology of Random 2-Complexes

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 π1(Y) is free and H2(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.

Random Simplicial Complexes

In this paper we propose a model of random simplicial complexes with randomness in all dimensions. We start with a set of n vertices and retain each of them with probability p0; on the next step we connect every pair of retained vertices by an edge with probability p1, and then fill in every triangle in the obtained random graph with probability p2, and so on. As the result we obtain a random simplicial complex depending on the set of probability parameters (\(p_{0},p_{1},\ldots,p_{r}\)), 0 ≤ p i ≤ 1. The multi-parameter random simplicial complex includes both Linial-Meshulam and random clique complexes as special cases. Topological and geometric properties of this random simplicial complex depend on the whole set of parameters and their thresholds can be understood as convex subsets and not as single numbers as in all the previously studied models. We mainly focus on foundations and on containment properties of our multi-parameter random simplicial complexes. One may associate to any finite simplicial complex S a reduced density domain \(\tilde{\mu }(S) \subset \mathbf{R}^{r}\) (a convex domain) which fully controls information about the values of the multi-parameter for which the random complex contains S as a simplicial subcomplex. We also analyse balanced simplicial complexes and give positive and negative examples. We apply these results to describe dimension of a random simplicial complex.

Geometry and topology of random 2-complexes

We study random 2-dimensional complexes in the Linial-Meshulam model and prove that the fundamental group of a random 2-complex Y has cohomological dimension ≤ 2 if the probability parameter satisfies p ≪ n−3/5. Besides, for

Large random simplicial complexes, I

{n^{ - 3/5}} \ll p \ll {n^{ - 1/2 - \epsilon }}

Topics of Stochastic Algebraic Topology

the fundamental group π1(Y) has elements of order two and is of infinite cohomological dimension. We also prove that for

Homological Domination in Large Random Simplicial Complexes

p \ll {n^{ - 1/2 - \epsilon }}