Topology of the octonionic flag manifold

Details Topology of the octonionic flag manifold Topology of the octonionic flag manifold

2008-09-25

arxiv.org/abs/0809.4318v2

Mathematics

Algebraic Topology

Version 2: exposition improved; two appendices added; 41 pages and 1 figure

Scientific paper

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.

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