Volumen Actual: Vol 28 (2024) No. 1

The ideal-valued index of fibrations with total space a \(G_2\) flag manifold

Noé Bárcenas, Jaime Calles Loperena.

Abstract:

By 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 comple 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 embedding between \(G_2/SO(4)\) and \(\tilde{G}_3(\mathbb{R}^7)\), the map that takes the orthogonal complement of a subspace, and the fiber bundle \(G_2/U(2)_\pm\rightarrow G_2SO(4)\). Here \(s\gamma\) means the associated sphere bundle of the bundle \(\gamma\). Furthermore, we derive a general formula for the \(n\)-fold producto bundle \(s\gamma^n\) for which we make the same computations. We finish our work with an application of our computations to a problem concerning discre geometry.

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Sistemas de cuadros rígidos: Aspectos topológico-algebraicos en fases y sus transiciones

Omar Alvarado-Garduño, Jesús González

Abstract:

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