Integral homology of real flag manifolds and loop spaces of symmetric spaces
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.
scientific article; zbMATH DE number 745274