Compact homogeneous spaces and their generalizations.

*(English. Russian original)*Zbl 1171.53037
J. Math. Sci., New York 153, No. 6, 763-798 (2008); translation from Sovrem. Mat., Fundam. Napravl. 22, 38-72 (2007).

This article is a survey on compact homogeneous spaces of the following types: a) homogeous spaces of compact Lie groups, b) compact solvable manifolds and c) compact homogeneous of the form \(G/\Gamma\) (homogeneous space with discrete stationary group).

We quote the last paragraph of the preface of the article for the content: “In section 1, some very important foundational material on homogeneous spaces is presented. The Tits-Onishchik bundles are introduced. In section 2, natural bundles for compact homogeneous spaces are introduced and studied. They are based on the action of the maximal compact subgroup of a transitive Lie group. Section 3 is devoted to some other bundles that are useful for describing the structure of compact homogeneous spaces. In section 4, principal bundles are studied, which correspond to natural bundles. A special class of homogeneous spaces is distinguished, for which the natural bundle is principal. In section 5 we single out natural classes of compact homogeneous spaces, both classical as well as relatively new: aspherical, solvable and semisimple, and some other homogeneous spaces. In section 6, some relations are considered related to homotopy groups of compact homogeneous spaces. Furthermore, the problem of computing real cohomologies of compact homogeneous spaces is discussed in detail. Section 7 is devoted to generalizations of the notion of compact homogeneous space, to quasicompact, and with greater detail, to plesiocompact homogeneous spaces. In section 8, several isolated special subjects are considered related to compact homogeneous spaces. We also formulate several unsolved problems in the theory of compact homogeneous spaces.”

We quote the last paragraph of the preface of the article for the content: “In section 1, some very important foundational material on homogeneous spaces is presented. The Tits-Onishchik bundles are introduced. In section 2, natural bundles for compact homogeneous spaces are introduced and studied. They are based on the action of the maximal compact subgroup of a transitive Lie group. Section 3 is devoted to some other bundles that are useful for describing the structure of compact homogeneous spaces. In section 4, principal bundles are studied, which correspond to natural bundles. A special class of homogeneous spaces is distinguished, for which the natural bundle is principal. In section 5 we single out natural classes of compact homogeneous spaces, both classical as well as relatively new: aspherical, solvable and semisimple, and some other homogeneous spaces. In section 6, some relations are considered related to homotopy groups of compact homogeneous spaces. Furthermore, the problem of computing real cohomologies of compact homogeneous spaces is discussed in detail. Section 7 is devoted to generalizations of the notion of compact homogeneous space, to quasicompact, and with greater detail, to plesiocompact homogeneous spaces. In section 8, several isolated special subjects are considered related to compact homogeneous spaces. We also formulate several unsolved problems in the theory of compact homogeneous spaces.”

Reviewer: Jing-Song Huang (Kowloon)

##### MSC:

53C30 | Differential geometry of homogeneous manifolds |

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\textit{V. V. Gorbatsevich}, J. Math. Sci., New York 153, No. 6, 763--798 (2008; Zbl 1171.53037); translation from Sovrem. Mat., Fundam. Napravl. 22, 38--72 (2007)

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