The ideal-valued index of fibrations with total space a $G_{2}$ flag manifold
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| author | Bárcenas, Noé Loperena, Jaime Calles |
| author_facet | Bárcenas, Noé Loperena, Jaime Calles |
| contents | 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 $γ$ over $\widetilde{G}_{4}(\mathbb{R}^{7})$, we compute the $\mathbb{Z}_2$ Fadell-Husseini index of the pullback bundle of $sγ$ 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 $sγ$ means the associated sphere bundle of $γ$. Furthermore, we derive a general formula for the $n$-fold product bundle $sγ^n$ for which we make the same computations. We finish our work with an application of our computations in a problem concerning discrete geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_01279 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The ideal-valued index of fibrations with total space a $G_{2}$ flag manifold Bárcenas, Noé Loperena, Jaime Calles Algebraic Topology 55N91 (Primary), 55T10, 57R20, 14M15 (Secondary) 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 $γ$ over $\widetilde{G}_{4}(\mathbb{R}^{7})$, we compute the $\mathbb{Z}_2$ Fadell-Husseini index of the pullback bundle of $sγ$ 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 $sγ$ means the associated sphere bundle of $γ$. Furthermore, we derive a general formula for the $n$-fold product bundle $sγ^n$ for which we make the same computations. We finish our work with an application of our computations in a problem concerning discrete geometry. |
| title | The ideal-valued index of fibrations with total space a $G_{2}$ flag manifold |
| topic | Algebraic Topology 55N91 (Primary), 55T10, 57R20, 14M15 (Secondary) |
| url | https://arxiv.org/abs/2103.01279 |