Some calculations of homotopy groups of symmetric spaces

Abstract

Introduction. We calculate the first few unstable homotopy groups of the symmetric spaces F,n = S02n1/U and Xn = U2,/Sp,, and of Sp.. The homotopy groups of Fn are needed in studying the existence of complex structures and knowledge of the first unstable group 7t2n 1(F,) is used in a paper of W. S. Massey [6]; in fact it was Professor Massey who first suggested to us the calculation of t2n_ 1(-X) for n 0 O (mod 4) (the other three parities of n are worked out by him), and suggested to us the use of some fibrations involving IF., or X., and spheres. Similarly, Xn is connected with almost quaternion structures. We rely heavily on Kervaire's calculations [4]. The space X,, possesses an involution a, induced by the involutory automorphism of U2, leaving Sp,, fixed. This automorphism of U2, extends to an inner automorphism of S041, and so induces a map a of period two on F2n. We also study the effect of a on homotopy groups; this is useful information, as shown in [2; 3]. The results are summarized in the following tables (the precise definition of a and other notation will be given following the tables): The groups 7r2n+r(rn):