Cellular homology of real flag manifolds. (English) Zbl 1426.57052
Let \(\mathbb{F}_\Theta= G/P_\Theta\) be a generalized flag manifold, where \(G\) is a real non-compact semi-simple Lie group and \(P_\Theta\) a parabolic subgroup. A classical result says that the Schubert cells, which are the closure of the Bruhat cells, endow \(\mathbb{F}_\Theta\) with a cellular \(CW\)-structure.
The authors exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in the Euclidean space \(\mathbb{R}^n\) and use them to compute the boundary operator \(\partial\) for the cellular homology. Clearly the cellular and the Morse-Witten complexes are intimately related since the Bruhat cells are the unstable manifolds of the gradient flow of a Morse function (see [J. J. Duistermaat et al., Compos. Math. 49, 309–398 (1983; Zbl 0524.43008)]). The result obtained by R. R. Kocherlakota [Adv. Math. 110, No. 1, 1–46 (1995; Zbl 0832.22020)] in the realm of Morse Homology that the coefficients of \(\partial\) are \(0\) or \(\pm 2\) is recovered. But, the formula given here is more refined in the sense that the ambiguity of signals in the Morse-Witten complex is solved.
The authors exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in the Euclidean space \(\mathbb{R}^n\) and use them to compute the boundary operator \(\partial\) for the cellular homology. Clearly the cellular and the Morse-Witten complexes are intimately related since the Bruhat cells are the unstable manifolds of the gradient flow of a Morse function (see [J. J. Duistermaat et al., Compos. Math. 49, 309–398 (1983; Zbl 0524.43008)]). The result obtained by R. R. Kocherlakota [Adv. Math. 110, No. 1, 1–46 (1995; Zbl 0832.22020)] in the realm of Morse Homology that the coefficients of \(\partial\) are \(0\) or \(\pm 2\) is recovered. But, the formula given here is more refined in the sense that the ambiguity of signals in the Morse-Witten complex is solved.
Reviewer: Marek Golasiński (Olsztyn)
MSC:
| 57T15 | Homology and cohomology of homogeneous spaces of Lie groups |
| 14M17 | Homogeneous spaces and generalizations |
| 32M10 | Homogeneous complex manifolds |
| 53C30 | Differential geometry of homogeneous manifolds |
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