The asphericity of random 2-dimensional complexes

We study random 2-dimensional complexes in the Linial - Meshulam model and prove that for the probability parameter satisfying $$p\ll n^{-46\mathrm{/}47}$$ 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 $Z^{\mathrm{\prime }}\subset Z$ is aspherical. This implies that $Y$ is homotopy equivalent to a wedge $Z\vee S^2\vee \mathrm{.}\mathrm{.}\mathrm{.}\vee S^2$ where $Z$ is a 2-dimensional aspherical simplicial complex. We also show that under the assumptions $$c\mathrm{/}n<p<n^{-1+ϵ},$$ where $c>3$ and $0<ϵ<1\mathrm{/}47$, 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.