Cellular Homology of Real Flag Manifolds

2018/10/01 by Lonardo Rabelo, Rabelo, Lonardo, Luiz A. B. San Martín +1 · 1 citation
Computer Science · Mathematics · #14M15 #57T15 #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1810.00934

openalex publication_date 2018/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbF_Θ=G/P_Θ be a generalized flag manifold, where G is a real noncompact semi-simple Lie group and P_Θ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow \mathbbF_Θ 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.

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