We describe the cohomological structure of the sequence of Heisenberg nilmanifolds M H 2 n - 1 → M H 2 n + 1 , n = 1, 2,... , in terms of bigraded symplectic structures on the tori T2n, the corresponding s l 2 -representations on the cohomology of T2n, and the Buchstaber spectral sequence (Bss).