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≪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'⊂Z is aspherical. This implies that Y is homotopy equivalent to a wedge Z∨S2∨..∨S2 where Z is a 2-dimensional aspherical simplicial complex. We also show that under the assumptions c/n<p<n-1+ε{lunate}, where c > 3 and 0<ε{lunate}<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 π1(Y) of a random 2-complex equals 2.