Cohomology theory of Lie groups and Lie algebras

Abstract

The present paper lays no claim to deep originality. Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to algebraic questions concerning Lie algebras('). This reduction proceeds in three steps: (1) replacing questions on homology groups by questions on differential forms. This is accomplished by de Rham's theorems(2) (which, incidentally, seem to have been conjectured by Cartan for this very purpose); (2) replacing the consideration of arbitrary differential forms by that of invariant differential forms: this is accomplished by using invariant integration on the group manifold; (3) replacing the consideration of invariant differential forms by that of alternating multilinear forms on the Lie algebra of the group. We study here the question not only of the topological nature of the whole group, but also of the manifolds on which the group operates. Chapter I is concerned essentially with step 2 of the list above (step 1 depending here, as in the case of the whole group, on de Rham's theorems). Besides considering invariant forms, we also introduce forms, defined in terms of a suitable linear representation of the group; Theorem 2.2 states that, when this representation does not contain the trivial representation, equivariant forms are of no use for topology; however, it states this negative result in the form of a positive property of equivariant forms which is of interest by itself, since it is the key to Levi's theorem (cf. later). Chapter II is concerned with step 3 of the above list. It is then necessary to assume that the group operates transitively on the manifold under consideration, that is, that this manifold is a homogeneous space relative to the group. Theorem 13.1, in connection with Theorem 2.3, indicates a method by which the Betti numbers of any homogeneous space attached to a connected compact Lie group may be computed algebraically. However, applications of this theorem are still lacking. In particular, it is desirable to obtain an algebraic proof of Samelson's theorem(3) to the effect that, if a closed subgroup