Geodesic
De Rham 2-Cohomology of Real Flag Manifolds
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\mathbb{F}_{\Theta }=G/P_{\Theta }$ be a flag manifold associated to a non-compact real simple Lie group $G$ and the parabolic subgroup $P_{\Theta }$. This is a closed subgroup of $G$ determined by a subset $\Theta $ of simple restricted roots of $\mathfrak{g}=\operatorname{Lie}(G)$. This paper computes the second de Rham cohomology group of $\mathbb{F}_\Theta$. We prove that it is zero in general, with some rare exceptions. When it is non-zero, we give a basis of $H^2(\mathbb{F}_\Theta,\mathbb{R})$ through the Weil construction of closed 2-forms as characteristic forms of principal fiber bundles. The starting point is the computation of the second homology group of $\mathbb{F}_{\Theta }$ with coefficients in a ring $R$.
Keywords: flag manifold, cellular homology, de Rham cohomology, characteristic classes.
Mots-clés : Schubert cell 
@article{SIGMA_2019_15_a50,
     author = {Viviana del Barco and Luiz Antonio Barrera San Martin},
     title = {De {Rham} {2-Cohomology} of {Real} {Flag} {Manifolds}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2019},
     volume = {15},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a50/}
}
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Viviana del Barco; Luiz Antonio Barrera San Martin. De Rham 2-Cohomology of Real Flag Manifolds. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a50/

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