Let 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 that 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].
The author thanks A. L. Onishchik for valuable remarks. 1. Homogeneous spaces and root systems Definition 1. Let G and G' be connected Lie groups, H and H' be closed subgroups of G and G', respectively, M = G/H, M' = G'/H'. The homogeneous space M is called isomorphic to M' if there exist an isomorphism f: G --G' and a diffeomorphism F: M -* M' such that F(g(m)) = f(g) (F(m)) for every g e G and m E M. Definition 2. Let G be a connected Lie group, H be a closed subgroup of G. The homogeneous space G/H is called almost effective if H does not contain non-discrete normal subgroups of G. Definition 3. Let G and G' be connected Lie groups, H and H' be closed subgroups of G and G ', respectively. The homogeneous space G/H is called an extension of G'/H' (in the strong sense) if there exists a monomorphismf: G' -m G such thatj(H') = f(G') n H and G = f(G') H. Definition 4.
Let k and K' be compact connected simply connected Lie groups, L and L' be connected subgroups of maximal rank of K and K', N and N' be central subgroups of K and K', respectively. Put K = K/N, K' = K'/N', L = LIN, L' = L'/N'. The homo- geneous space M = K/L is called almost isomorphic to M' = K'/L', M -M', if the homogeneous space K/L is isomorphic to k'/L'.