The integral cohomology ring of E8/T

Abstract

We determine the integral cohomology ring of the homogeneous space E 8 /T 1 ·E 7 by the Borel presentation and a method due to Toda. Then using the Gysin exact sequence associated with the circle bundle S 1 → E 8 /E 7 → E 8 /T 1 ·E 7 , we also determine the integral cohomology of E 8 /E 7 .

The Chow Rings of Generalized Grassmannians

“…In 1974, H. Toda [32] initiated the project computing the integral cohomology of homogeneous spaces G/H with G an exceptional Lie group and H ⊂ G a torsionfree subgroup of maximal rank. This amounts to combining Borel's method [3] with the previous results on H * (G; Z p ) (as a module over the Steenrod algebra) for all primes p. After Toda, the cohomologies of the G/H considered in Theorems 1, 3-6 have been studied by Toda, Watanabe, Ishitoya, and Nakagawa in [23,24,28,[33][34][35] in which the generators are specified only by their degrees.…”

“…They utilized various spectral sequence techniques for certain fibrations associated with G/H [1,3,21,32,36]. However, these techniques encounter the same difficulties when applied to Lie groups G with torsion [23,24,28,[32][33][34][35], in particular, when G is not prime to an exceptional Lie group (see discussion in Sect. 7.1).…”

“…In particular, Nakagawa recently announced a presentation for the ring H * (E 8 /E 7 · S 1 ) in [29].…”

On the Cohomology of Some Exceptional Symmetric Spaces

“…In fact, the rings of invariants of the Weyl groups W (E 7 ), W (E 8 ) have been computed in [26], [16], [17]. They read:…”

Schubert presentation of the cohomology ring of flag manifolds

“…In [20] Similarly, for the adjoint Lie groups P G with G = SU (n), Sp(n), E 6 and E 7 m, one has (see Remark 6.5. For the earlier works studying the presentation of the ring H * (G/T ), see [3,5,27,29,30,33,34]. A basic requirement of intersection theory [19] is to present the cohomology H * (X) of a projective variety X by explicit described geometrical cycles, such as the Schubert classes of flag manifolds, so that the intersection multiplicities can be computed by the cup products on the ring H * (X).…”