On Homogeneous Cohomology Spheres

Abstract

A space is said to be a homogeneous integral cohomology k-sphere if it is a coset space G/H of a compact connected Lie group G by a closed (but not necessarily connected) subgroup Hand if HI(G/H; Z) = HI(Sk; Z) for all i. These spaces have recently come to be of some importance in the theory of transformation groups. When they arise in the study of general (nondifferentiable) transformation groups of a cohomology manifold (or of a manifold) it is usually not evident whether or not they are simply connected, let alone whether or not they are spheres. However, Borel has shown that if a homogeneous integral cohomology sphere is simply connected, then it is a sphere (see [2, 4.6], and also [3]). This fact for the even-dimensional case was also proved by H. C. Wang. On the other hand, there is a well-known example of such a space which is not simply connected. This is the quotient of SO (3) by the icosahedral subgroup I. It might be expected that there would be many more exceptional cases besides SO(3)/I, however we shall show that this is not the case. The purpose of this paper, then, is to prove that any homogeneous integral cohomology sphere G/H either is simply connected, is a circle, or is homeomorphic (naturally) to SO (3)/I. Thus, using the result of Borel, we will see that SO(3)/I is the only homogeneous integral cohomology sphere which is not a sphere. We shall use the following notation. If M and N are spaces and L is a ring then we say that M -, N if HI(M; L) HI(N; L). Z denotes the ring of integers and R denotes the field of rational numbers. If K is a compact Lie group, we denote the identity component of K by K0. We also let NG(K) be the normalizer of K in G, and Z0(K), the centralizer of K in G. We drop the subscript G when this causes no confusion. Aut (K) denotes the automorphism group of K and Int (K) denotes the group of inner automorphisms of K. If G, and G, are Lie groups then G1o G2 denotes a group of the form (G, x G2)/N, where N is a finite normal subgroup of G1 x G2, and similarly with any number of factors. That is,