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We prove sharper versions of theorems of Linial-Meshulam and Meshulam-Wallachwhich describe the behavior for (Z/2)-cohomology of a random k-dimensionalsimplicial complex within a narrow transition window. In particular, we showthat within this window the (k-1)st Betti number is in the limit Poissondistributed. For k=2 we also prove that in an accompanying growth process, withhigh probability, first cohomology vanishes exactly at the moment when the lastisolated (k-1)-simplex gets covered by a k-simplex.

Inside the critical window for cohomology of random k-complexes