Sharp vanishing thresholds for cohomology of random flag complexes

• 成果类型: Article
• 署名作者: Kahle, Matthew
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2014.179.3.5
• 发表日期: 2014
• 页码: 1085-1107
• 关键词: homological connectivity

摘要

For every k >= 1, the k-th cohomology group H-k (X,Q) of the random flag complex X similar to X (n,p) passes through two phase transitions: one where it appears and one where it vanishes. We describe the vanishing threshold and show that it is sharp. Using the same spectral methods, we also find a sharp threshold for the fundamental group pi(1)(X) to have Kazhdan's property (T). Combining with earlier results, we obtain as a corollary that for every k >= 3, there is a regime in which the random flag complex is rationally homotopy equivalent to a bouquet of k-dimensional spheres.