Cellular homology of real flag manifolds

Abstract

Let FΘ=G∕PΘ be a generalized flag manifold, where G is a real non-compact semi-simple Lie group and PΘ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow FΘ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in Rn and use them to compute the boundary operator ∂ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of ∂ are 0 or ±2 (so that Z2-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.