The ideal-valued index of fibrations with total space a $G_2$ flag manifold

Abstract

Using the cohomology of the $G_2$-flag manifolds $G_2/U(2{)}_{\pm }$, and their structure as a fiber bundle over the homogeneous space $G_2/SO(4)$, we compute the ${\mathbb{Z}}_2$ Fadell-Husseini index of such fiber bundles, for the ${\mathbb{Z}}_2$ action given by complex conjugation. Also, considering the tautological bundle $\gamma$ over ${\tilde{G}}_4({\mathbb{R}}^7)$, we compute the ${\mathbb{Z}}_2$ Fadell-Husseini index of the pullback bundle of $s\gamma$ along the composition of the fiber bundle $G_2/U(2{)}_{\pm }\to G_2/SO(4)$, the embedding between $G_2/SO(4)$ and ${\tilde{G}}_3({\mathbb{R}}^7)$, and the map that takes the orthogonal complement of a subspace. Here $s\gamma$ means the associated sphere bundle of $\gamma$. Furthermore, we derive a general formula for the $n$-fold product bundle $s{\gamma }^n$ for which we make the same computations. We finish our work with an application of our computations in a problem concerning discrete geometry.

21 pages, 6 figures