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 / 47a 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 ∨ S 2 ∨ … ∨ S 2where Z is a 2‐dimensional aspherical simplicial complex. We also show that under the assumptions c / n < p < n − 1 + ϵ , where c > 3 and 0 < ϵ < 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 groupπ 1 ( Y ) of a random 2‐complex equals 2. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 261–273, 2015