The asphericity of random 2-dimensional complexes

TL;DR

This paper investigates the topological properties of random 2-dimensional complexes in the Linial-Meshulam model, showing that under certain probability conditions, these complexes contain aspherical subcomplexes satisfying the Whitehead conjecture, with implications for their homology and fundamental groups.

Contribution

It proves the existence of aspherical subcomplexes satisfying the Whitehead conjecture in random 2-complexes for specific probability ranges, advancing understanding of their topology.

Findings

Existence of disjoint tetrahedra leading to aspherical subcomplexes

Subcomplexes satisfy the Whitehead conjecture, ensuring asphericity of all subcomplexes

Fundamental group has cohomological dimension 2 in certain probability regimes

Abstract

We study random 2-dimensional complexes in the Linial - Meshulam model and prove that for the probability parameter satisfying $$p\ll n^{-46/47}$$ a random 2-complex $Y$ contains several pairwise disjoint tetrahedra such that the 2-complex $Z$ obtained by removing any face from each of these tetrahedra is aspherical. Moreover, we prove that the obtained complex $Z$ satisfies the Whitehead conjecture, i.e. any subcomplex $Z'\subset Z$ is aspherical. This implies that $Y$ is homotopy equivalent to a wedge $Z\vee S^2\vee...\vee S^2$ where $Z$ is a 2-dimensional aspherical simplicial complex. We also show that under the assumptions $$c/n<p<n^{-1+\epsilon},$$ where $c>3$ and $0<\epsilon<1/47$, the complex $Z$ is genuinely 2-dimensional and in particular, it has sizable 2-dimensional homology; it follows that in the indicated range of the probability parameter $p$ the cohomological dimension of the fundamental group $\pi_1(Y)$ of a random 2-complex equals 2.