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Topology of Lie groups, I and II. Transl. from the Jap. by Mamoru Mimura and Hirosi Toda. (English) Zbl 0757.57001

Translations of Mathematical Monographs. 91. Providence, RI: American Mathematical Society (AMS). iv, 451 p. (1991).
The purpose of this work is to present a fairly complete and self- contained account of the homology, cohomology, and homotopy of Lie groups and associated spaces.
In the first part, the authors restrict themselves to the classical groups and take advantage of the methods of linear algebra to study the geometric-topological aspects of these groups and their classical homogeneous spaces. Then they use the methods of algebraic topology rather extensively to complete this part of their program. They finish the first part with a proof of the Bott periodicity theorems (without using Morse theory).
In the second volume of the work, they turn to the general case of compact connected Lie groups. Here the geometric-topological problems are studied by means of representation theory and, using Morse theory, they give an account of the Bott-Samuelson results on symmetric spaces and their loop spaces. Finally, they focus on the study of the cohomology of the exceptional groups.

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57Txx Homology and homotopy of topological groups and related structures
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