Let K be a compact connected Lie group, L be a 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 K/L is positive. The homotopy classification of such homogeneous spaces KIL in case L is connected was obtained in . We prove that two compact connected homogeneous spaces KIL and K'/L' of positive Euler characteristic are homotopy equivalent if and only if they are diffeomorphic. This result was announced in .
The author thanks A. L. Onishchik for valuable remarks.
1. Homogeneous Spaces and Root Systems
The present paper is a continuation of . We use the notations of this paper. Let us recall that if K is a compact connected semisimple Lie group and L its connected subgroup of maximal rank, then the homogeneous space KIL is defined uniquely up to an "almost isomorphism" by a pair (R, U), where R is the root system of K and U a regular subsystem. Any such pair (R, U) defines the graded ring H*(K/L, Z). it is proved that non-isomorphic pairs (R, U) and (R', U') define, as a rule, non-isomorphic graded rings H*(K/L, Z) and H*(K'/L', Z), respectively, and all exceptions are listed (we assume that the homogeneous spaces KIL and K'/L' are almost effective). We may generalize the concept of an almost isomorphy of homogeneous spaces (cf. Definition 4 on p. 119 in ) to the case of non-connected isotropy subgroups omitting the word "connected" in line 2 of this definition. (Also correct the misprints: read "K and K"' instead of "K and K"' in the lines 2 and 3 of this definition.) There exists a bijection of the set of classes of almost isomorphic compact connected homogeneous spaces of positive Euler characteristic onto the set of isomorphic triplets (R, U, A), where R is a root system, U a regular subsystem and A a group, defined in terms of the Weyl group W(R).
Let us give the precise definitions. Let R be a root system and U a regular subsystem of R. Denote by Aut (R) the group of automorphisms of the root system R and let Aut (R, U) and W(R, U) be the normalizers of the subgroup W(U) in the groups Aut (R) and W(R), respectively. Put