Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 051, 23 pages      arXiv:1811.07854      https://doi.org/10.3842/SIGMA.2019.051

De Rham 2-Cohomology of Real Flag Manifolds

Viviana del Barco ^ab and Luiz Antonio Barrera San Martin ^b
a) UNR-CONICET, Rosario, Argentina
^b) IMECC-UNICAMP, Campinas, Brazil

Received January 08, 2019, in final form June 25, 2019; Published online July 05, 2019

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 $R$.

Key words: flag manifold; cellular homology; Schubert cell; de Rham cohomology; characteristic classes.

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References

1. Bott R., Samelson H., Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964-1029.
2. Hatcher A., Algebraic topology, Cambridge University Press, Cambridge, 2002.
3. Helgason S., Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, Inc., New York - London, 1978.
4. Knapp A.W., Lie groups beyond an introduction, Progress in Mathematics, Vol. 140, Birkhäuser Boston, Inc., Boston, MA, 1996.
5. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. II, Interscience Tracts in Pure and Applied Mathematics, Vol. 15, Interscience Publishers John Wiley & Sons, Inc., New York - London - Sydney, 1969.
6. Kocherlakota R.R., Integral homology of real flag manifolds and loop spaces of symmetric spaces, Adv. Math. 110 (1995), 1-46.
7. Mare A.-L., Equivariant cohomology of real flag manifolds, Differential Geom. Appl. 24 (2006), 223-229, arXiv:math.DG/0404369.
8. Rabelo L., San Martin L.A.B., Cellular homology of real flag manifolds, arXiv:1810.00934.
9. San Martin L.A.B., Álgebras de Lie, 2nd ed., UNICAMP, Campinas, 2010.
10. Silva J.L., Rabelo L., Half-shifted Young diagrams and homology of real Grassmannians, arXiv:1604.02177.
11. Warner G., Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Vol. 188, Springer-Verlag, New York - Heidelberg, 1972.
12. Wiggerman M., The fundamental group of a real flag manifold, Indag. Math. (N.S.) 9 (1998), 141-153.