The asphericity of random 2‐dimensional complexes

Abstract

AbstractWe study random 2‐dimensional complexes in the Linial–Meshulam model and prove that for the probability parameter satisfying urn:x-wiley:10429832:media:rsa20499:rsa20499-math-0002 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 is aspherical. This implies that Y is homotopy equivalent to a wedge where Z is a 2‐dimensional aspherical simplicial complex. We also show that under the assumptions urn:x-wiley:10429832:media:rsa20499:rsa20499-math-0005 where c > 3 and , 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 of a random 2‐complex equals 2. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 261–273, 2015