which was conjectured by Serre [2, p. 428]. The theory of [8] showed that the groups in (3) had isomorphic p-primary components for p > 2n, and so we show that this actually holds for p > 2 (it does not hold for p = 2). We prove (1) and (2) by showing that, although the above fibrations do not have a cross-section, there is a map which behaves like a cross-section in homology with coefficients any field of characteristic = 2. If G = su(2n), K = sp(n) or if G = su(2n + 1), K = so(2n + 1) and if a is the automorphism of G leaving K fixed (complex conjugation in the second case, complex conjugation followed by an inner automorphism in the first) then the map in question is the map q: G/K , G given by q(gK) = g(g)-1. Although this map q has been noted previously in the theory of symmetric spaces, its action on cohomology has been determined only recently (in our note 15], which considers general symmetric and other homogeneous spaces). In a later paper we will show that this map is the generalization to symmetric spaces of the characteristic maps for fibre bundles over spheres considered in [9, ?? 23, 24]. We also show that the direct sum decompositions (1) and (2) are simply the decompositions into the +1 and -1 eigenspaces of the map induced