The asphericity of random 2‐dimensional complexes. Issue 2 (12th April 2013) - Costa, A.E.; Farber, M.
Abstract We study random 2‐dimensional complexes in the Linial–Meshulam model and prove that for the probability parameter satisfying <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp15" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />p≪n−46/47</alternatives> 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 <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp0m" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Z′⊂Z</alternatives> is aspherical. This implies that Y is homotopy equivalent to a wedge <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp38" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Z∨S2∨…∨S2</alternatives> where Z is a 2‐dimensional aspherical simplicial complex. We also show that under the assumptions <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp2q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />c/n<p<n−1+ϵ, </alternatives> where c > 3 and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp5c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />0<ϵ<1/47</alternatives>, 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 <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp4t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />π1(Y)</alternatives> of a random 2‐complex equals 2. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 261–273, 2015