Topology of the octonionic flag manifold

TL;DR

This paper explores the topological and equivariant cohomology and K-theory of the octonionic flag manifold, providing explicit descriptions and new computational tools for these sophisticated geometric structures.

Contribution

It offers Borel type descriptions of cohomology and K-theory for the octonionic flag manifold, including Goresky-Kottwitz-MacPherson type formulas, advancing understanding of its equivariant topology.

Findings

Borel type descriptions of $Spin(8)$-equivariant cohomology

GKM type description of the cohomology ring

GKM type description of the equivariant K-theory ring

Abstract

The octonionic flag manifold $Fl(\mathbb{O})$ is the space of all pairs in $\mathbb{O}P^2\times \mathbb{O}P^2$ (where $\mathbb{O}P^2$ denotes the octonionic projective plane) which satisfy a certain "incidence" relation. It comes equipped with the projections $\pi_1,\pi_2 : Fl(\mathbb{O})\to \mathbb{O}P^2$, which are $\mathbb{O}P^1$ bundles, as well as with an action of the group $Spin(8)$. The first two results of this paper give Borel type descriptions of the usual, respectively $Spin(8)$-equivariant cohomology of $Fl(\mathbb{O})$ in terms of $\pi_1$ and $\pi_2$ (actually the Euler classes of the tangent spaces to the fibers of $\pi_1$, respectively $\pi_2$, which are rank 8 vector bundles on $Fl(\mathbb{O})$). Then we obtain a Goresky-Kottwitz-MacPherson type description of the ring $H^*_{Spin(8)}(Fl(\mathbb{O}))$. Finally, we consider the $Spin(8)$-equivariant $K$-theory ring of $Fl(\mathbb{O})$ and obtain a Goresky-Kottwitz-MacPherson type description of this ring.