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 SIGMA, 2019, Volume 15, 051, 23 pages (Mi sigma1487)

De Rham 2-Cohomology of Real Flag Manifolds

Viviana del Barcoab, Luiz Antonio Barrera San Martina

a IMECC-UNICAMP, Campinas, Brazil
b UNR-CONICET, Rosario, Argentina

Abstract: 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, Schubert cell, de Rham cohomology, characteristic classes.

 Funding Agency Grant Number Fundação de Amparo à Pesquisa do Estado de São Paulo 2015/23896-52017/13725-42012/18780-0 National Council for Scientific and Technological Development (CNPq) 476024/2012-9 V. del Barco supported by FAPESP grants 2015/23896-5 and 2017/13725-4. L.A.B. San Martin supported by CNPq grant 476024/2012-9 and FAPESP grant 2012/18780-0.

DOI: https://doi.org/10.3842/SIGMA.2019.051

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Bibliographic databases:

ArXiv: 1811.07854
MSC: 57T15, 14M15
Received: January 8, 2019; in final form June 25, 2019; Published online July 5, 2019
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Citation: Viviana del Barco, Luiz Antonio Barrera San Martin, “De Rham 2-Cohomology of Real Flag Manifolds”, SIGMA, 15 (2019), 051, 23 pp.

Citation in format AMSBIB
\Bibitem{DelSan19} \by Viviana~del Barco, Luiz Antonio Barrera~San Martin \paper De Rham 2-Cohomology of Real Flag Manifolds \jour SIGMA \yr 2019 \vol 15 \papernumber 051 \totalpages 23 \mathnet{http://mi.mathnet.ru/sigma1487} \crossref{https://doi.org/10.3842/SIGMA.2019.051} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000474630200001}