Topology of transitive transformation groups. (English) Zbl 0796.57001

Onishchik, Arkadi L.

Leipzig: Johann Ambrosius Barth. xv, 300 p. (1994).

The book is devoted to the topology of compact Lie groups and homogeneous spaces and its applications to the solution of the following fundamental problem: classify all transitive actions of compact Lie groups on a given homogeneous space and determine all inclusions among them. The latter problem reduces to determination of all factorizations \(G = AH\) of any connected compact Lie group \(G\) into the product of two Lie subgroups \(A\), \(H\). In the first introductory chapter, basic results of the theory of Lie groups and homogeneous spaces are presented. They include the classification of compact Lie groups and the description of their representations, parabolic subgroups and subgroups of maximal rank.
Chapter 2 contains the basic facts about graded algebra and differential graded algebra, including Sullivan’s theory of minimal models and an algebraic version of Hopf and Samelson theorem about real cohomology of a Lie group.
Chapter 3 treats the real cohomology of compact Lie groups and their homogeneous spaces. The theory of Weil algebras is demonstrated, and the fundamental Cartan theorem is proved. The theorem establishes the isomorphism between the cohomology algebra of a homogeneous space \(G/H\) and the cohomology of the Cartan algebra associated with the pair \((G,H)\). A minimal model of a homogeneous space \(M\) is described and other homotopy invariants of \(M\) are constructed. They are applied in Chapter 4 for the classification of the factorization of connected compact Lie groups and the determination of the full automorphism groups of complex flag manifolds and some Riemannian homogeneous spaces.
In Chapter 5 certain classification results for compact homogeneous spaces are given including the classification of transitive actions of Lie groups on sphere and flag manifolds and the classification of homogeneous spaces of positive Euler characteristic.

Reviewer: D.V.Alekseevsky (Moskva)

MSC:

57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57T15	Homology and cohomology of homogeneous spaces of Lie groups
57S25	Groups acting on specific manifolds
53C30	Differential geometry of homogeneous manifolds
22-02	Research exposition (monographs, survey articles) pertaining to topological groups
22E15	General properties and structure of real Lie groups
57T20	Homotopy groups of topological groups and homogeneous spaces
57T10	Homology and cohomology of Lie groups

Keywords:

transitive actions of compact Lie groups on homogeneous space; compact Lie groups; homogeneous spaces; representations; parabolic subgroups; subgroups of maximal rank; minimal models; real cohomology of a Lie group; Weil algebras; Cartan algebra; factorization of connected compact Lie groups; automorphism groups of complex flag manifolds; homogeneous spaces of positive Euler characteristic