The integral cohomology of varieties endowed with the cup product becomes a lattice. Torelli theorems or classification of automorphisms are among the important motivations for studying the cohomology lattices. After a general introduction on these concepts, we will focus on the generalized Kummer 4-fold K_2(T) and its quotient by a symplectic involution K_2(T)/i. Following our main goal to provide a describtion of the integral cohomology of these two varieties, we will also show some general techniques to calculate the integral cohomology of quotient varieties.