Rational cohomology of the Rosenfeld projective planes

Question

The bioctonionic plane $(\mathbb{C} \otimes \mathbb{O})\mathbb{P}^2$, the quarteroctonionic plane $(\mathbb{H} \otimes \mathbb{O})\mathbb{P}^2$ and the octooctonionic plane $(\mathbb{O} \otimes \mathbb{O})\mathbb{P}^2$ are manifolds of dimension 32, 64 and 128 with isometry groups $E_6$, $E_7$ and $E_8$ respectively. They are not actual projective planes but can be defined most elegantly as quotients of $E_6$, $E_7$ and $E_8$ by a subgroup, see for an exposition John Baez's website. I'm interested in properties of these manifolds. Has anyone calculated their rational (or even integral) cohomology? Are they frameable?

Answer 1

The spaces you mention are symmetric spaces. The notation from Cartan's list is:

• $E\operatorname{III}$ for the quotient of $E_6$

• $E\operatorname{VI}$ for the quotient of $E_7$

• $E\operatorname{VIII}$ for the quotient of $E_8$.

Computations for $E\operatorname{III}$ have been done in

• H. Toda and T. Watanabe. The integral cohomology ring of $F_4/T$ and $E_6/T$. J. Math. Kyoto Univ 14 (1974), 257-286.

Computations for $E\operatorname{VI}$ have been done in

• M. Nakagawa. The mod 2 cohomology ring of the symmetric space $E\operatorname{VI}$. J. Math. Kyoto Univ. 41 (2001), 535-556.

I don't know of integral cohomology computations for the last one, this is more difficult due to the presence of torsion. Generally, there are a lot of computations of cohomology of Lie groups and their homogeneous spaces by the Japanese school. See e.g. Topology of Lie groups I and II by Mimura-Toda.

If you are interested in the rational cohomology, then Borel gave a presentation for the cohomology rings for homogeneous spaces $G/H$ where $H\hookrightarrow G$ where $H$ has maximal rank in $G$ (if I'm not mistaken this should apply to all the cases):

• A. Borel. Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. Math. 57 (1953), pp. 115-207.