The ideal-valued index of fibrations with total space a $G_{2}$ flag manifold

TL;DR

This paper computes the $bZ_2$ Fadell-Husseini index for certain $G_2$-flag manifold fiber bundles, providing new formulas and an application to discrete geometry.

Contribution

It introduces explicit index computations for $G_2$-flag manifold bundles and their products, extending the understanding of their topological invariants.

Findings

Computed the $bZ_2$ Fadell-Husseini index for $G_2/U(2)_{ ext{±}}$ bundles.

Derived a general formula for the index of $sg^n$ bundles.

Applied the index computations to a problem in discrete geometry.

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 $\widetilde{G}_{4}(\mathbb{R}^{7})$, 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 $\widetilde{G}_{3}(\mathbb{R}^{7})$, 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.