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

Noé Bárcenas, Jaime Calles Loperena

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 $γ$ 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.

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

Abstract

Using the cohomology of the -flag manifolds , and their structure as a fiber bundle over the homogeneous space , we compute the Fadell-Husseini index of such fiber bundles, for the action given by complex conjugation. Also, considering the tautological bundle over , we compute the Fadell-Husseini index of the pullback bundle of along the composition of the fiber bundle , the embedding between and , and the map that takes the orthogonal complement of a subspace. Here means the associated sphere bundle of . Furthermore, we derive a general formula for the -fold product bundle for which we make the same computations. We finish our work with an application of our computations in a problem concerning discrete geometry.

Paper Structure

This paper contains 21 sections, 20 theorems, 84 equations, 5 figures.

Table of Contents

  1. Introduction.
  2. The exceptional Lie group $G_{2}$.
  3. Octonions algebra
  4. The exceptional Lie group $G_{2}$: three equivalent definitions
  5. $G_{2}$ flag manifolds
  6. The space of associative subspaces $G_{2}/SO(4)$.
  7. The six dimensional sphere $G_{2}/SU(3)$.
  8. The space of complex coassociative $2$-planes $G_{2}/U(2)_{+}$.
  9. The space of complex non-coassociative $\mathbf{2}$-planes $\mathbf{G_{2}/ U(2)_{-}}$.
  10. The full flag manifold $\mathbf{G_{2}/ U(1)\times U(1)}$.
  11. Calculations of Fadell-Husseini Index.
  12. The Fadell-Husseini Index
  13. Dold's argument
  14. The Fadell-Husseini index of $\rho_1$ and $\rho_2$.
  15. The Fadell-Husseini index of the pullback bundle $\zeta_n$
  16. ...and 6 more sections

Key Result

Theorem \ref{thm: BorelCohom G2/U2 & FH rho1&2}

The Fadell-Husseini index of $\rho_1$ and $\rho_2$ is given by where $H^*(B{\mathbb Z}_2; {\mathbb F}_2 ) = {\mathbb F}_2[t]$ with $\deg(t) = 1$. Consequently, the Borel cohomology of $G_{2}/U(2)_{\pm}$ is given by

Figures (5)

  • Figure 1: $E_2^{p,q} = H^p ( S^6 ;H^q( \mathbb{CP}^2 ;{\mathbb F}_2 ) ) \Rightarrow H^*( G_2/U(2)_- ;{\mathbb F}_2 )$.
  • Figure 2: $E_2^{p,q} = H^p ( G_2/U(2)_{-} ;H^q( S^2 ;{\mathbb F}_2 ) ) \Rightarrow H^*( G_{2}/U(1) \times U(1);{\mathbb F}_2 )$.
  • Figure 3: $E_2^{p,q} = H^p ( G_2/SO(4) ;H^q( S^2 ;{\mathbb F}_2 ) ) \Rightarrow H^*( G_2/U(2)_{+};{\mathbb F}_2 )$.
  • Figure 4: $E_4^{p,q} = E_2^{p,q} = H^p ( B{\mathbb Z}_2 \times G_2/U(2)_{\pm} ;H^q( S^3 ;{\mathbb F}_2) )$
  • Figure 5: Morphism of spectral sequences induced by $p_k$.

Theorems & Definitions (35)

  • Theorem \ref{thm: BorelCohom G2/U2 & FH rho1&2}
  • Theorem \ref{thm: FH Pullback}
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Definition 3.5
  • ...and 25 more