Integral homology of real flag manifolds and loop spaces of symmetric spaces
type: Item
instance of: scholarly article
title: Integral homology of real flag manifolds and loop spaces of symmetric spaces
zbMATH Open document ID: 0832.22020
author: R. R. Kocherlakota
published in: Advances in Mathematics
publication date: 1996-03-07T00:00:00Z
review text: A generalization is presented of an earlier work [\textit{R. Bott} and \textit{H. Samelson}, Sympos. Int. Topologia Algebraica 282-284 (1958; Zbl 0101.397)] in which the so called real flag manifolds (RFM) for compact Lie groups with involution have been introduced and the related integral homology group for the simplest case \(\mathbb{Z} = \mathbb{Z}_2\) has been computed. The generalized RFM are considered as orbits (invariants) which are generated when \(K\) acts on \(\mathfrak g\). Here \(K\) is the Lie group which is defined as the fixed set of symmetric pairs \((G,\tau) = G^\tau\) where \(G\) is an initial compact simply connected Lie group and \(\tau : G \to G\) an involution to which there corresponds an involution \(\tau_* : {\mathfrak g} \to {\mathfrak g}\) of the related Lie algebra \(\mathfrak g\) of \(G\). By definition, the algebra \(\mathfrak g\) is the direct sum \({\mathfrak g} = {\mathfrak p} \oplus {\mathfrak k}\), where \(\mathfrak p\) and \(\mathfrak k\) (Lie algebra of \(K\)) are the \(+1\) and \(-1\) eigenspaces for \(\tau_*\), respectively. An algorithm is presented for the computation of the integral homology groups \(H_* (K_\eta, \mathbb{Z})\) for any \(\eta \in {\mathfrak p}\) and arbitrary \(\mathbb{Z}\) of generalized RFM and of based loop spaces \(\Omega(G/K)\) of infinite dimension of compact symmetric spaces. By using the root system of the underlying symmetric spaces this algorithm reduces the computation to combinatorial calculation. As a tool to understand the geometry of the spaces under consideration some theorems on boundary maps are formulated and proved by using Morse theory of the moment map. The general investigation is illustrated by several concrete examples.
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 57T10
Mathematics Subject Classification ID: 37D15
Mathematics Subject Classification ID: 55P35
zbMATH DE Number: 745274
zbMATH Keywords: Bott-Samelson scheme
zbMATH Keywords: real flag manifolds
zbMATH Keywords: compact Lie groups
zbMATH Keywords: involution
zbMATH Keywords: orbits
zbMATH Keywords: compact simply connected Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: homology groups
zbMATH Keywords: loop spaces
zbMATH Keywords: Morse theory
reviewed by: A. A. Bogush
MaRDI profile type: Publication
full work available at URL: aima.1995.1001
OpenAlex ID: W2045696125
DOI: 10.1006/AIMA.1995.1001
description: scientific article; zbMATH DE number 745274