On the Cohomology of Compact Homogeneous Spaces of Nilpotent Lie Groups

As part of the general problem of investigating the topological structure of homogeneous spaces of non-compact Lie groups, A. Malcev [7]2 obtained the following results. Any homogeneous space of a connected nilpotent Lie group is the topological product of an euclidean space and a compact space which is itself a homogeneous space of a certain connected nilpotent Lie group. If the homogeneous space is compact, it can be expressed in the form 9I = 3/T where 5 is a simply connected nilpotent Lie group and Z a discrete subgroup. Recently, Y. Matsushima [8] has proved that the cohomology groups H*(9) (with real coefficients) of the homogeneous space 9I are isomorphic with the cohomology groups H* (g) of the Lie algebra g of 5 for dimensions r = 1 and 2. The first and main purpose of the present paper is to establish the isomorphism of these cohomology groups for every dimension r. Indeed, we shall prove Theorem 1 which gives a canonical isomorphism of the cohomology algebras H*(9)) and H*(g). As corollaries we obtain a few results on the Euler characteristic and Betti numbers of compact homogeneous spaces of nilpotent Lie groups, which are similar to known results concerning homogeneous spaces of compact Lie groups. It also follows that a homogeneous space of a connected nilpotent Lie group is always orientable. For the homogeneous spaces of compact Lie groups we have the well known theory of invariant integrals of E. Cartan, namely, the cohomology of a homogeneous space of a compact Lie group can be obtained from the complex of invariant differential forms on it (for example, [4], Chapter I, Theorem 2.3). The second purpose of this paper is to show that, in the notation 9 = = / as above, the complex of invariant differential forms on 91 is isomorphic with the complex of invariant cochains of the Lie algebra g (Theorem 2). Then we shall give a very simple example to the effect that the theory of invariant integrals does no longer hold for compact homogeneous spaces of non-compact Lie groups. Such examples do not seem to have been known. For the proof of Theorem 1 we need some fundamental concepts and results in the cohomology theory of principal fiber bundles as well as the theory of differential sequences in the sense of Leray and Koszul. In ?2 we shall give those concepts which are necessary for our purpose and prove a lemma from which Theorem 1 can be easily derived. The formulation and proof of Theorem 1 to-