Symmetric Spaces Which Are Real Cohomology Spheres

This is a survey in which we collate some known results using semi-standard techniques, dropping the condition of simple connectivity in Kostant's work [2] and proving Theorem 1. Let M be a compact connected riemannian symmetric space. Then M is a real cohomology (dim M)-sphere if and only if (1) M is an odd dimensional sphere or real projective space; or (2) M = M/Γ where (a) M = S 2 r i X •. X STM, rt > 0, product of m > 1 even dimensional spheres, and (b) Γ consists of all γ = γλX X γm, where γ{ is the identity map or the antipodal map of S 2 r , and the number of yt which are antipodal maps, is even or (3) M = SU(3)/SO(3) orM = {SU(3)/Z3}/SO(3) or (4) M = O(5)/O(2) X O(3), non-oriented real grassmannian of 2-planes through 0 in R. In (2) we note πx(M) = Γ = {Ίj^ ~\ in particular the even dimensional spheres are the case m — 1. In (3) we note that the first case is the universal 3-fold covering of the second case. In (4) we have πλ{M) = Z2. Theorem 1 is based on a series of lemmas which can be pushed, with appropriate modification, to the case of a real cohomology w-sphere of dimension greater than n. Here we make the convention that a 0-sphere is a single point. By using a cohomology theory which satisfies the homotopy axiom (such as singular theory) we can also drop the requirement of compactness. Thus we push the method of proof of Theorem 1 and obtain Theorem 2. Let M be a connected riemannian symmetric space. Then M is a real cohomology n-sphere, 0 < n < dim M, if and only if M = MΎ.M" where (a) M" is a product whose I > 0 factors are euclidean spaces and irreducible symmetric spaces of noncompact type, and (β) M is one of the following spaces. (1) W = M/Γ, where M = S r i X X STM is the product of m > 0 spheres of positive even dimensions 2ru Γ=(Z2) m consists of all γλX Xγm such that γi is the identity or antipodal map on S s θ is any one of the 2 characters on Γ, and Γ is the kernel of θ. Express θ = θiχ θi$9 where