Cellular Homology of Real Flag Manifolds

TL;DR

This paper explicitly parametrizes Schubert cells in real flag manifolds, computes their cellular homology, and clarifies the boundary coefficients, enhancing understanding of their topological structure.

Contribution

It provides explicit parametrizations of Schubert cells by closed balls and computes the boundary operator, refining previous Morse homology results.

Findings

Boundary coefficients are 0 or ±2.

Z2-homology is freely generated by cells.

Explicit cell parametrizations are constructed.

Abstract

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.