Inside the critical window for cohomology of random k ‐complexes

We prove sharper versions of theorems of Linial–Meshulam and Meshulam–Wallach which describe the behavior for ( ℤ / 2 ) ‐cohomology of a random k ‐dimensional simplicial complex within a narrow transition window. In particular, we show that if Y is a random k ‐dimensional simplicial complex with each k ‐simplex appearing i.i.d. with probability p = k log n + c n , with k ≥ 1 and c ∈ ℝ fixed, then the dimension of cohomologyβ k − 1 ( Y ) is asymptotically Poisson distributed with meane − c / k ! . In the k = 2 case we also prove that in an accompanying growth process, with high probability,H k − 1 ( Y , ℤ / 2 ) vanishes exactly at the moment when the last ( k − 1 ) ‐simplex gets covered by a k ‐simplex, a higher‐dimensional analogue of a “stopping time” theorem about connectivity of random graphs due to Bollobás and Thomason. Random Struct. Alg., 2015 © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 102–124, 2016