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 $(\mathbb{Z}/2)$-cohomology of a random $k$-dimensional simplicial complex within a narrow transition window. In particular, we show that within this window the Betti number $\beta^{k-1}$ is in the limit Poisson distributed. For $k=2$ we also prove that in an accompanying growth process, with high probability, first cohomology vanishes exactly at the moment when the last isolated $(k-1)$-simplex gets covered by a $k$-simplex.