We follow the computations in [2], [5], and [8] to deduce the cohomology rings of various Lie groups (SU(n), U(n), Sp(n) with Z-coefficients and SO(n), Spin(n), G2, F4, E6, E7, E8 with field coefficients) using the Serre spectral sequence.

Let G be a semisimple algebraic group defined over the rational numbers, K a maximal compact subgroup of G = G(R), and Γ ⊂ G(Q) a neat arithmetic subgroup. LetX = Γ\G/K be the locally symmetric space associated to Γ, and E the local system on X constructed out of a finite-dimensional, irreducible, algebraic representation E of G. Fix a maximally Q-split torus S in G; S is assumed to be nontrivi...