Let $\mathbb{ F}_{ \Theta }=G/P_{ \Theta }$ be a generalized flag manifold, where $G$ is a real noncompact semi-simple Lie group and $P_{ \Theta }$ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow $\mathbb{ F}_{ \Theta}$ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in $\mathbb{ R}^{ n}$ and use them to compute the boundary operator $\partial $ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of $\partial $ are $0$ or $\pm 2$ (so that $\mathbb{ Z}_{ 2}$-homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse-Witten complex is solved.
Categories: math.AT math.DG