  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.
