Le K be a compact connected Lie group, L be a connected closed subgroup of K. It is well known that L is a subgroup of maximal rank of K if and only if the Euler characteristic of the manifold M = K/L is positive. Such homogeneous spaces M have been classified in [7, 10]. However, their topological classification was unknown. This classification is obtained in the present article. We show tha two compact homogeneous spaces M = K/L and M′ = K′/L′ of positive Euler characteristic are diffeomorphic if and only if the graded rings H *(M,Z) and H *(M′,Z) are isomorphic. We also obtain the rational homotopy classification of such homogeneous spaces which is not equivalent to the differential one. These results were announced in [15].
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Shchetinin, A. On a class of compact homogeneous spaces I. Ann Glob Anal Geom 6, 119–140 (1988). https://doi.org/10.1007/BF00133035
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DOI: https://doi.org/10.1007/BF00133035