The asphericity of random 2‐dimensional complexes. Issue 2 (12th April 2013) - Costa, A.E.
Abstract 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, 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 group π1(Y) of a random 2‐complex equals 2. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 261–273, 2015