On the cohomology of some exceptional symmetric spaces

TL;DR

This survey explores the construction of an octonionic Kähler 8-form on exceptional symmetric spaces, linking Clifford structures to cohomology and primitive Betti numbers, advancing understanding of their topological invariants.

Contribution

It introduces a canonical octonionic Kähler 8-form construction and relates Clifford structures to cohomology and Betti numbers on exceptional symmetric spaces.

Findings

Constructed a canonical octonionic Kähler 8-form.

Linked Clifford structures to primitive Betti numbers.

Described cohomology generators for Cayley-Rosenfeld planes.

Abstract

This is a survey on the construction of a canonical or "octonionic K\"ahler" 8-form, representing one of the generators of the cohomology of the four Cayley-Rosenfeld projective planes. The construction, in terms of the associated even Clifford structures, draws a parallel with that of the quaternion K\"ahler 4-form. We point out how these notions allow to describe the primitive Betti numbers with respect to different even Clifford structures, on most of the exceptional symmetric spaces of compact type.